{"id":"e859edd7-1ae3-42eb-b3ac-ca5352b1c7dd","arxiv_id":"2509.06632","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For thick rotating spherical shells and large Prandtl numbers, no-slip outer boundaries lower the critical Rayleigh number relative to stress-free boundaries, and internal heating moves onset away from the tangent cylinder.","lead":"This paper maps how the choice of boundary conditions (sticky versus slippery walls) and the way heat is supplied change the earliest onset of swirling convection in rotating spherical shells, using numerical stability calculations. It finds that slippery outer boundaries do not always make convection easier: for thick shells and high Prandtl numbers, sticky walls actually trigger convection at lower heating.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Continuation-restricted m-minimization is untested near mode jumps; a missed global azimuthal mode could flip the boundary-condition preference transitions.","rationale":"The reader identified the same load-bearing assumption: the continuation strategy constrains the search over m to a small neighbourhood of the previous optimum, which is not stress-tested near mode jumps. This is a real methodological soft spot because the paper's headline conclusions are comparisons of critical values that are sometimes very close, e.g., the boundary-condition preference inversions in Figures 5 and 10. If the true global mode changes by more than the neighbourhood size, the reported Rac for one boundary condition could be a local minimum, and the ordering RaNS_c versus RaSF_c could be reversed. The paper otherwise has good support: the spectral discretization is documented, resolution convergence is shown in Appendix B, and selected critical values are validated against the literature. However, those validations do not cover the transition regions where mode jumps are most likely. I do not see a stronger competing concern. The Ekman-layer 'destabilising' interpretation is an inference without a direct boundary-layer diagnostic, but the ordering of critical Rayleigh numbers would remain valid even if the mechanism were mislabelled. The moderate-Ekman-number limitation is explicitly acknowledged and does not undermine the parameter study as stated. Therefore my concern is the same as the reader's, and it does not change the verdict: the paper should remain CONDITIONAL pending an unconstrained m-search at the transition points.","tokens_in":24266,"tokens_out":4678,"duration_ms":48989,"concrete_test":"Reproduce the critical curves at selected transition points using an exhaustive m-sweep: fix (E,Pr,χ) at the reported inversion locations, including Pr≈1.7 and Pr≈0.9 at χ=0.35 for differential and internal heating, and χ≈0.45–0.76 at E=3×10−5 for internal heating with Pr=1. For each point, solve γ(Ra,m)=0 for every m from 1 to Mmax (set Mmax about 20% above the largest reported mc) using neutral or randomly initialized eigenpairs, without seeding from the previously converged mc. Then compare the global argmin Rac(m) with the reported value. If any reported mc differs, or if the sign of 1−RaSF_c/RaNS_c changes under the exhaustive sweep, the continuation restriction is responsible and the preference transitions are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central comparisons rest on global minima over the azimuthal wavenumber m, but Section 2.2 states that 'the minimisation over m is then restricted to a small neighbourhood of the previously identified mc'. Near the reported mode transitions, such as the kink at χ≈0.4 in the internally heated case and the sign changes of 1−RaSF_c/RaNS_c in Figure 10, the true minimizer may jump by more than that neighbourhood. If a jump is missed, the reported Rac would be a local, not global, minimum for one boundary condition, directly biasing the relative difference that defines the preferred boundary condition. The paper does not report any test that restarts the m-search from an unconstrained set or verifies the global minimum at these transition points. Since several preference inversions occur where RaNS_c and RaSF_c are close, even a small m-caused error could flip their ordering. This is a methodological premise, not a claim about the physics, and it is the weakest point in the argument because the headline conclusions depend on it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the linear onset of thermal convection in rotating spherical shells using a spectral method in spherical harmonics and Chebyshev polynomials. It systematically varies mechanical boundary conditions (no-slip, stress-free, and mixed with no-slip inner/stress-free outer) and heating modes (differential versus internal) over Pr ∈ [0.1, 100], χ ∈ [0.2, 0.8], and E ∈ {10^-4, 3×10^-5, 10^-5}. The main claims are that the preferred mechanical boundary condition at onset is not universal: for sufficiently thick shells or large Pr the no-slip outer boundary gives a lower critical Rayleigh number than the stress-free boundary, attributed to a destabilising Ekman layer; internal heating generally raises Rac, shifts onset to larger azimuthal wavenumbers and frequencies, and moves the critical column away from the tangent cylinder; and mixed boundary conditions with no-slip inner/stress-free outer closely mimic the purely stress-free case. The paper also analyses how the choice of nondimensionalisation can reverse apparent trends of Rac with χ, and provides fits for scaling laws.","tokens_in":24433,"tokens_out":5148,"duration_ms":42826,"significance":"The study is potentially valuable because it maps, over a wide parameter range, where the conventional choice of mechanical boundary condition in rotating-shell convection models changes the onset threshold. If correct, the results imply that the preferred boundary condition is a function of Pr, χ, E, and heating type rather than a fixed modelling convention, with direct implications for nonlinear convection and dynamo simulations. The numerical work is quantitatively anchored by comparisons with several prior studies (Dormy et al. 2004, Al-Shamali et al. 2004, Barik et al. 2023, Fan et al. 2024) and by convergence tests showing Rac converged to within 0.003% at the most demanding parameters (Appendix B); the data are openly available. The principal weakness is methodological: the global minimisation over the azimuthal wavenumber m relies on a continuation search restricted to a small neighbourhood of the previous m_c, and this is not tested at the reported mode transitions, where the boundary-condition preference reversals are decided by small differences in Rac.","major_comments":[{"comment":"The global critical mode is defined as Rac = min_m Ra_c^(m), but the implementation restricts the minimisation over m to 'a small neighbourhood of the previously identified mc' during continuation. Near reported transitions—for example the kink at χ≈0.4 in the internally heated case (Section 3.2.2, Figure 9b) and the sign changes of 1−Ra_c^SF/Ra_c^NS in Figure 10—the true minimizer may jump azimuthal wavenumber by more than that neighbourhood, so the reported Rac could be a local rather than global minimum. Because the preferred-boundary-condition conclusion is determined by the sign of a small difference between Ra_c^NS and Ra_c^SF, a missed global mode at these points could flip the ordering. The paper does not report any test that restarts the m-search from an unconstrained set or otherwise verifies the global minimum at these parameter values; the authors should add such a test and state whether the preference conclusions survive it.","section":"Section 2.2"},{"comment":"The mechanistic claim that the Ekman boundary layer at the outer boundary 'becomes destabilising' for sufficiently thick shells or large Pr is inferred solely from the inequality Ra_c^NS < Ra_c^SF. No direct computation or scaling analysis of the Ekman layer's contribution (e.g., boundary-layer suction or viscous dissipation) is provided, so the mechanism is an interpretation rather than an established result. The observed ordering of critical Rayleigh numbers is a robust numerical finding, but the causal language in the abstract overstates what the calculations show; either add a supporting boundary-layer argument or rephrase the claim as 'consistent with a destabilising role of the Ekman layer' rather than asserting it.","section":"Abstract, Section 3.1.1, Section 4"}],"minor_comments":[{"comment":"Equation (11) defines C1 = χ^{-1}Rac, but the caption of Table 3 states C1 = Rac; the notation should be made consistent.","section":"Section 3.2, Scaling laws and Table 3"},{"comment":"'radius radio' should read 'radius ratio'.","section":"Section 3.2, Flow patterns"},{"comment":"'whether or nor such a situation' contains a typo ('nor' should be 'not').","section":"Section 3.1.1"},{"comment":"The heading and text use 'nondimsionalisation' and 'nondimensionalistaion'; these should be corrected to 'nondimensionalisation'.","section":"Section 4, Remarks on nondimsionalisation"},{"comment":"The text notes that frequency rescaling is needed when the timescale changes, but no explicit conversion formula for ω is given; adding the conversion to Table 1 would improve usability.","section":"Table 1 and Section 2.3"}],"recommendation":"major_revision","confidential_remarks":"The continuation-restricted m-minimization concern is legitimate and bears directly on the headline boundary-condition preference results; it should be resolved with explicit unconstrained m sweeps at the transition points. The rest of the numerical validation is solid, and the paper is within the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a careful, well-executed parameter study. It gives the community something useful: a systematic map of how the preferred mechanical boundary condition at onset depends on Prandtl number, radius ratio, Ekman number, and heating mode. Previous papers covered pieces of this space; this one spans Pr from 0.1 to 100 and chi from 0.2 to 0.8 for both differential and internal heating, with no-slip, stress-free, and mixed boundaries. The finding that no-slip boundaries can destabilise onset for thick shells or large Pr (and the companion map for internal heating) is a genuinely useful guide for people choosing boundary conditions in nonlinear simulations.\n\nThe numerics look solid. The authors validate against Dormy et al., Barik et al., Al-Shamali et al., Fan et al., and Christensen and Aubert, with suitable conversions between nondimensionalisations. They document resolution convergence to 0.003% and root-finding tolerances. The data are openly available. The section on how the choice of Rayleigh-number definition changes the trend of Rac with chi is a good reminder that 'easier onset' claims need to be tied to a specific nondimensionalisation.\n\nWhere I would push back: the continuation method restricts the azimuthal-wavenumber search to a small neighbourhood of the previous critical m. Near the reported transitions (the kink around chi=0.4 in the internally heated case, and the sign changes in 1-RaSF/RaNS in Figure 10), the true global mode can jump by more than that neighbourhood. The paper does not report any check that restarts the search from an unrestricted set at those points. Since the boundary-condition preference is defined by the difference between two near-threshold values, a missed mode jump could flip the ordering. I don't think this invalidates the broad picture, but it deserves a robustness test before the transition lines are taken as quantitative.\n\nTwo smaller caveats. The claim that the Ekman layer becomes destabilising is inferred from the ranking of critical Rayleigh numbers, not from a direct boundary-layer diagnostic; that is a reasonable reading but not a demonstration. And the Ekman numbers are moderate (1e-4 to 1e-5), which the authors acknowledge, so the relevance to planetary interiors is suggestive rather than direct.\n\nOverall: this is a solid, honest paper and a good candidate for refereeing. I would ask the authors to add a small set of restart tests on the m-minimization near the transitions, and to soften or better support the 'destabilising Ekman layer' phrasing. With that, it would be a useful reference and a good data source.","headline":"A careful, well-validated linear-stability scan mapping how boundary-condition preference depends on geometry and heating; the continuation-restricted m-search deserves a robustness check near transitions.","tokens_in":25013,"tokens_out":2970,"would_cite":true,"duration_ms":25949,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"No-slip boundaries trigger convection before stress-free ones in thick rotating shells or at high Prandtl numbers.","keywords":["rotating spherical shells","thermal convection","boundary conditions","Ekman boundary layer","internal heating","critical Rayleigh number","Prandtl number","columnar convection"],"falsifier":"Re-run the linear stability search at parameter values straddling the reported transitions, for example with Prandtl number near 1.7 in differential heating and radius ratio near 0.4 or inside the boundary-condition reversal region in internal heating at Ekman number $3\\times10^{-5}$, scanning all azimuthal wavenumbers from several independent starting guesses instead of a small neighbourhood of the previous critical mode; any mismatch would show the reported critical Rayleigh number is not the global value.","tokens_in":24010,"feed_emoji":"🌀","tokens_out":7094,"duration_ms":61981,"temperature":0.7,"pith_summary":"This paper asks which mechanical boundary conditions and which heating mechanism make thermal convection easiest to start in a rotating spherical shell, the canonical model for fluid motion in planetary cores and stellar interiors. The central finding is that the preferred boundary condition is not a fixed modelling choice: for thick shells or large Prandtl numbers, the Ekman boundary layer at the outer surface becomes destabilising, so no-slip walls give a lower critical Rayleigh number than stress-free walls, while the reverse holds elsewhere. It also finds that internal heating, compared with differential heating, raises the critical Rayleigh number, selects larger azimuthal wavenumbers and oscillation frequencies, and moves the first convective columns away from the tangent cylinder. If these results hold, numerical models of planetary and stellar convection need to justify their boundary-condition and heating choices rather than treating either as a neutral convention.","feed_headline":"Thick shells flip which boundary wins at convection onset","feed_subtitle":"Ekman boundary layers turn destabilizing, so no-slip beats stress-free; internal heating pushes onset outward.","key_machinery":"The argument is carried by the linear stability problem of the rotating-shell equations, solved by a spectral expansion in Chebyshev polynomials and spherical harmonics and by a continuation in parameter space that tracks the marginal mode with the largest growth rate. The physical mechanism invoked is the Ekman boundary layer, whose thickness scales as $E^{1/2}$: it restrains the flow at low Prandtl number but becomes destabilising for thick shells or large Prandtl numbers, reversing which boundary condition lowers the critical Rayleigh number. A second structural element is the critical cylinder radius, located by the maximum of the azimuthally averaged radial velocity in the equatorial plane; internal heating detaches the onset from the tangent cylinder and moves that radius with Prandtl number and Ekman number.","core_discovery":"At linear onset in a Boussinesq rotating spherical shell at moderate Ekman numbers ($10^{-4}$ to $10^{-5}$), the critical Rayleigh number, azimuthal wavenumber, and oscillation frequency are computed for Prandtl numbers 0.1 to 100, radius ratios 0.2 to 0.8, differential and internal heating, and no-slip, stress-free, and mixed boundary conditions. The paper's main discovery is that the outer Ekman boundary layer can be either stabilising or destabilising: under differential heating, no-slip boundaries give a lower critical Rayleigh number than stress-free boundaries once the Prandtl number exceeds about 1.7, and under internal heating the transition occurs near a Prandtl number of about 0.9 and also appears as a function of radius ratio, with the relative difference between the two critical values changing sign in an Ekman-number-dependent way. Internal heating raises the critical Rayleigh number by up to an order of magnitude, increases the wavenumber and frequency at onset, and places the critical convection column at a radius that departs from the previously predicted asymptotic value when the Prandtl number differs from 1. Mixed boundaries with no-slip on the inner sphere and stress-free outside closely reproduce the purely stress-free results, because the convection columns interact mainly with the outer surface.","pith_inferences":["If the outer boundary dominates onset, then for stars modelled with mixed boundary conditions the stress-free approximation may also hold in weakly nonlinear regimes, though whether it survives into strongly nonlinear dynamo states is untested.","The Prandtl-number drift of the critical radius found here suggests the asymptotic global theory's fixed value is only the unit-Prandtl-number case; a Prandtl-dependent correction to that theory would be a natural next step.","Because the trend of the critical Rayleigh number with radius ratio reverses when the outer radius rather than the shell thickness sets the length scale, comparisons of convective ease between planets with different inner-core sizes should use the outer-radius scaling to avoid a purely geometric artefact.","A combined heating model with both differential and internal sources would likely interpolate between the two preference maps, and the boundary-condition transition could then occur at intermediate Prandtl numbers and radius ratios."],"forward_implications":["Planetary and stellar convection models must treat no-slip versus stress-free boundaries as a physical parameter with a measurable effect, not a numerical convenience: at thick shells or high Prandtl numbers, no-slip onset is easier.","Models using no-slip inner and stress-free outer boundaries can, near onset, be approximated by purely stress-free conditions because the outer boundary dominates.","Internal heating stabilises the shell relative to differential heating, so planets or stars with strong volumetric heating require stronger thermal forcing to begin convection.","At Ekman numbers relevant to planetary interiors, the sign of the boundary-condition preference depends on shell thickness under internal heating, with no-slip favoured for small radius ratios and stress-free for large ones.","The reported critical values across the parameter grid provide a benchmark for nonlinear simulations, whose onset is expected to fall close to these linear critical parameters."],"supporting_citations":[{"why":"Supplies the baseline critical values and the global-theory critical radius for Prandtl number 1, as well as the earlier observation that internal heating with no-slip boundaries can lower the critical Rayleigh number.","marker":"[19]"},{"why":"Provides the earlier result that Ekman boundary layers influence rotating convection and that internal heating favours no-slip at larger Prandtl numbers, which this paper extends across parameter space.","marker":"[42]"},{"why":"Supplies the scaling-law forms used to fit the critical Rayleigh number, wavenumber, and frequency as functions of Prandtl number and radius ratio.","marker":"[7]"},{"why":"Validates the Prandtl-number dependence of the present critical values through comparison with recent spherical-shell calculations.","marker":"[10]"},{"why":"Validates the no-slip differential-heating results through comparison with dynamo-scaling calculations.","marker":"[12]"},{"why":"Provides the radius-ratio-dependent onset values used to validate the scans in shell thickness.","marker":"[31]"},{"why":"Supplies earlier radius-ratio results and a modified scaling law for comparison at thin and thick shells.","marker":"[34]"},{"why":"Establishes the global asymptotic theory for the critical cylinder in a full sphere, against which the internal-heating critical radius is measured.","marker":"[18]"}],"fun_headline_variants":["No-slip beats stress-free when Ekman layer destabilizes","Internal heating raises critical Rayleigh, shifts onset column","Thick shells flip boundary winner at convection onset","Outer boundary layer decides onset: no-slip favored at high Pr","Ekman boundary layer turns destabilizing for thick shells"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that the most unstable azimuthal wavenumber changes only gradually as parameters move in small steps; if the true minimum jumps abruptly between steps, the reported critical Rayleigh number could be a local rather than global minimum.","fun_headline_variants_meta":{"raw":{"variants":["No-slip beats stress-free when Ekman layer destabilizes","Internal heating raises critical Rayleigh, shifts onset column","Thick shells flip boundary winner at convection onset","Outer boundary layer decides onset: no-slip favored at high Pr","Ekman boundary layer turns destabilizing for thick shells"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000318,"raw_usage":{"total_tokens":1842,"prompt_tokens":1035,"completion_tokens":807,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":727}},"tokens_in":651,"tokens_out":807,"duration_ms":7092,"temperature":1.0,"reasoning_tokens":727,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:14:32.575630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the linear stability search at parameter values straddling the reported transitions, for example with Prandtl number near 1.7 in differential heating and radius ratio near 0.4 or inside the boundary-condition reversal region in internal heating at Ekman number $3\\times10^{-5}$, scanning all azimuthal wavenumbers from several independent starting guesses instead of a small neighbourhood of the previous critical mode; any mismatch would show the reported critical Rayleigh number is not the global value.","supporting_citations":[{"cited_title":"The influence of Ekman boundary layers on rotating convection","cited_arxiv_id":null,"evidence_quote":"Provides the earlier result that Ekman boundary layers influence rotating convection and that internal heating favours no-slip at larger Prandtl numbers, which this paper extends across parameter space."},{"cited_title":"Scaling behaviour of rotating convection in a spherical shell with different Prandtl numbers","cited_arxiv_id":null,"evidence_quote":"Validates the Prandtl-number dependence of the present critical values through comparison with recent spherical-shell calculations."},{"cited_title":"Onset of convection in rotating spherical shells: Variations with radius ratio","cited_arxiv_id":null,"evidence_quote":"Provides the radius-ratio-dependent onset values used to validate the scans in shell thickness."},{"cited_title":"Varying the spherical shell geometry in rotating thermal convection","cited_arxiv_id":null,"evidence_quote":"Supplies earlier radius-ratio results and a modified scaling law for comparison at thin and thick shells."},{"cited_title":"The onset of thermal convection in a rapidly rotating sphere","cited_arxiv_id":null,"evidence_quote":"Establishes the global asymptotic theory for the critical cylinder in a full sphere, against which the internal-heating critical radius is measured."}],"review_version":2}