Influence of Planetary Rotation on Supersonic Flow of Lava Planets: A Two-Dimensional Horizontal Model Analysis

Feng Ding; Wanying Kang; Zhuo-Yang Song

REVIEW 3 major objections 4 minor 36 references

Influence of Planetary Rotation on Supersonic Flow of Lava Planets: A Two-Dimensional Horizontal Model Analysis

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The rotation of a lava planet produces a Coriolis-driven circulation that breaks the substellar-point symmetry assumed by all earlier axisymmetric models, with the first-order asymmetry proportional to the nondimensional spin rate.

desk verdict A careful first-order perturbation of the lava-planet flow model shows rotation generates an asymmetric circulation, but the expansion is not uniformly small for the massive close-in planets that are the prime targets. read the letter →

arxiv 2501.04350 v2 pith:GUNO2LFK submitted 2025-01-08 astro-ph.EP

Zhuo-Yang Song , Feng Ding , Wanying Kang This is my paper · ORCID

classification astro-ph.EP

keywords lavaplanetsmineralvaporatmosphereplanetaryrotationCorioliseffectsupersonicflowperturbationexpansiontidallylockedRossbynumber

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 extends the standard one-dimensional model of supersonic mineral-vapor flow on lava planets to two horizontal dimensions by treating planetary rotation as a small perturbation. It shows that the Coriolis force deflects the day-to-night supersonic transport into an east-west circulation, so the atmosphere is no longer symmetric about the substellar-antistellar axis. The asymmetric component is the part of the flow that would show up in phase-curve observations, drive magma-ocean currents, and slowly deform the planet's shape, so knowing its structure matters for interpreting lava planets. The paper also maps how the asymmetry depends on planetary mass, density, surface temperature, and the sticking coefficient for surface-atmosphere exchange.

What carries the argument

The load-bearing object is the perturbation expansion in $\tilde{\omega}$, the ratio of planetary rotation speed to the sound speed scale $\sqrt{C_p T_0}$, which is the inverse Rossby number built on the sound speed. Because lava-planet winds are supersonic, the expansions of all fields are written as the known 1D axisymmetric solution plus a first-order term, and the azimuthal structure is fixed by the Coriolis force to be first harmonic in longitude: $u_1\cos\phi$ for the circulation and $V_1, P_1, T_1$ for transport, pressure, and temperature. Substituting this ansatz reduces the 2D problem to a linear system of four ODEs in the tidally locked latitude $\theta$, with coefficient matrices constructed entirely from zeroth-order quantities; the sonic-point singularity is handled by a binary search on the boundary condition at the substellar point so that the solution passes smoothly through the critical point.

What would settle it

Run a nonlinear three-dimensional atmospheric simulation of a lava planet with the standard parameters but with $\tilde{\omega}$ near 0.5: if the azimuthal wind and the $m=1$ pressure pattern do not scale approximately linearly with $\tilde{\omega}$ and match $u_1, V_1, P_1, T_1$, the perturbation expansion is falsified for realistic spin rates. A cheaper test is to compare the paper's fitted extrapolation for K2-141b with the observed phase curve: the model predicts a specific westward-convergence/eastward-divergence asymmetry, so a phase curve with the opposite longitudinal shift, or with a much larger amplitude than $\tilde{\omega}$ times the first-order solution, would rule the claim out.

Watch

Extended reading notes

Core claim

Starting from the established axisymmetric solution in which mineral vapor evaporates near the substellar point and accelerates to supersonic speeds toward the nightside, the paper solves the first-order correction to the steady horizontal flow produced by the Coriolis force. Expanding all fields in the nondimensional spin parameter $\tilde{\omega} = \omega a / \sqrt{C_p T_0}$, it finds that the first-order azimuthal (circulation) velocity takes the form $u^{(1)} = -\omega a\,u_1(\theta)\cos\phi$ while the transport, pressure, and temperature corrections vary as $\sin\phi$, with the amplitudes determined by a linear system of ODEs whose coefficients depend only on the zeroth-order axisymmetric state. The calculation confirms that rotation generically breaks substellar-point symmetry, producing convergence west of the antistellar point and divergence east of it, and that the nondimensional first-order state is universal for a given zeroth-order state, independent of the actual rotation rate. Relative to the background flow, the asymmetric corrections are largest for small, cool, low-mass lava planets and for weak surface sticking; for the standard simulation the azimuthal circulation reaches hundreds of meters per second.

Load-bearing premise

The whole calculation assumes the spin rate is a small parameter and that the first-order term captures the asymmetry; the paper itself notes that typical lava planets have Rossby number only about 2 and that the solver fails for massive planets like K2-141b, so this small-spin premise is not established for the most observationally interesting cases.

Editorial extensions

If this is right

The substellar-to-antistellar symmetry assumed by earlier models is broken by rotation; the first-order pattern is a zonal circulation that accelerates with latitude and is strongest away from the substellar point.
Because the nondimensional first-order state depends only on the zeroth-order state, one reference solution can be rescaled to any planetary rotation rate for which $\tilde{\omega}$ stays small.
Asymmetric pressure and temperature fields of order $\tilde{\omega}$ times the background imply observable signatures in thermal phase curves and in the surface wind stress that drives magma-ocean circulation.
The relative asymmetry is largest for small, cool lava planets and for surfaces with smaller sticking coefficient $\alpha$, which behaves like a reduced effective mass $M^* \equiv \alpha^{3/2}(\rho_p/\rho_\oplus)^{1/2} M_p$.
In the standard simulation the azimuthal circulation $\omega a u_1$ reaches hundreds of meters per second, meaning rotation cannot be neglected when interpreting the flow.

Reading between the lines

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

If the paper's exponential fits for the relative asymmetry continue to hold beyond the solved mass range, the asymmetry on a massive planet like K2-141b would be sizable even though the solver fails there; that extrapolation is proposed but not established.
The residual analysis shows wave-number-two terms that are small on the dayside but comparable to the first-order terms beyond $\theta \approx 130^\circ$, so a second-order calculation would likely shift the longitude of maximum pressure and temperature away from the pure $\sin\phi$ prediction.
Because the linear stability analysis finds growing modes only on the nightside, where the flow is supersonic and cannot propagate information upstream, the steady dayside solution may be a robust attractor even if the nightside is time-dependent.
The predicted surface wind-stress pattern gives a concrete forcing field for models of lava-ocean currents, offering a testable bridge between this atmospheric result and observed shape or thermal anomalies.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, the punchline: this is the first paper to add planetary rotation to the 1D supersonic-flow framework for lava planet mineral-vapor atmospheres. It does this by a first-order perturbation expansion in the nondimensional spin parameter, and it works out the asymmetric circulation that results. The mechanism—Coriolis deflection of the transport flow drives an azimuthal circulation that breaks substellar symmetry—is physically sensible and matches the pattern you'd expect from the shallow-water tidally-locked literature.

What's genuinely good: the derivation is transparent and mostly self-contained. The authors start from the known 1D axisymmetric solution, linearize the governing equations, and solve a well-posed set of ODEs. The appendices are a real strength. They check the residual of the truncated expansion (Appendix C), test sensitivity to the integration starting point (Appendix B), and attempt a crude linear stability analysis (Appendix D). That's more honesty than most papers in this area show. The approximation for the circulation velocity u1 is also nice: a simple balance between Coriolis acceleration and momentum drag from evaporation, which reproduces the exact solution away from the antistellar point.

Now the soft spots, and they are real but not fatal. The expansion parameter is 0.13 in the standard run, but the relevant Rossby number for typical lava planets is only about 2, so the expansion is not as small as you'd like. The paper's own residual analysis shows the second-order terms are comparable to the first-order Coriolis term on the nightside, and the stability appendix reports growing modes there. That doesn't kill the mechanism, but it means the quantitative solution is only reliable on the dayside and near the terminator. The bigger issue is extrapolation to massive planets like K2-141b. The solver fails there, so the authors fit exponential curves to the low-mass results and extrapolate. They are upfront that this is a hypothesis, but the extrapolated numbers should not be treated as predictions.

The paper doesn't ship code or data, which is a minor annoyance. The reference list includes several self-citations to Kang et al. (2021), but that's appropriate because the method is built directly on that model.

Bottom line: this is a solid, careful first step. The central mechanism is plausible and the framework is reusable. The quantitative claims for the most interesting planets are conditional on better convergence and a proper nonlinear treatment, so the broad applicability statements need to be softened. Send it to peer review; a good referee will push on the expansion validity and the extrapolation, but the core contribution deserves a forum.

Who it's for: exoplanet atmosphere modelers who care about lava planets, and anyone extending 1D condensation-driven flow models to include rotation. Worth engaging with seriously.

Referee Report

3 major / 4 minor

Summary. This paper develops a two-dimensional horizontal model for the mineral-vapor atmospheres of synchronously rotating lava planets. Starting from the axisymmetric one-dimensional solution of Kang et al. (2021), the authors expand the steady primitive equations in the nondimensional rotation rate tilde_omega and solve first-order ODEs (Eq. 12) for the longitude-asymmetric transport and circulation fields, assuming the sinusoidal longitude dependence of Eq. (11). The standard simulation (Mp = 0.05 M_sun, Ts0 = 3200 K, P = 18 h) has tilde_omega = 0.13 and yields asymmetric winds of order several hundred m/s. The paper also maps the sensitivity of first-order amplitudes to a modified mass M* and Ts0, and proposes exponential fits for massive planets where the direct solver fails.

Significance. If the linearized first-order solution is valid, the central claim that planetary rotation generates a substantial azimuthal circulation breaking substellar-point symmetry is physically plausible, and the nondimensional rescaling of the first-order state is a useful design principle for future models. The derivation is transparent and the residual and linear-stability checks in Appendices C and D are a commendable attempt to assess the truncation error. The main weakness is that the paper has not established quantitative validity of the first-order truncation for the nightside and for massive close-in planets such as K2-141b, which are among the most observationally relevant targets. The paper is a promising framework rather than a fully validated prediction.

major comments (3)

[Section 2.3, Eq. (9)] The perturbation expansion in tilde_omega is the load-bearing step, but the paper's own checks show that it is not uniformly valid. In the standard simulation, the residual λ22,2D becomes comparable to the first-order Coriolis term λ22,1D for θ ≥ 130° (Figure 8), and the linear stability analysis in Appendix D gives positive Re(σ) for θ > 80° (Figure 9). These are exactly the nightside regions where the paper describes convergence west and divergence east of the antistellar point. The authors should either include second-order contributions in the nightside or explicitly restrict the stated validity to θ ≲ 110° and explain why the nightside circulation pattern can still be trusted despite the residual being first-order-comparable.
[Section 3.3, Eq. (18) and Figure 5] The exponential extrapolation Δi = Ai exp(-Bi M*) + Ci is used to estimate first-order amplitudes for planets more massive than Earth, including K2-141b, but the fits are only over M*/M⊕ ≤ 0.25. The manuscript itself states that the solver fails for planets more massive than Earth, and no independent test of the extrapolation is provided. Since K2-141b is a key observational target mentioned in the Introduction, a fitted curve over a limited mass range cannot carry the quantitative applicability claim. Please provide a convergence check or an independent numerical method for the massive-planet regime, or explicitly reframe those results as a tentative conjecture rather than a prediction.
[Section 2.3, Eq. (11)] The assumed sinusoidal longitude dependence is introduced as a prior and is not derived from the first-order equations. While the Coriolis forcing has a cos φ form, the piecewise mass-flux terms in Eq. (6) and the condensation-boundary treatment in Appendix A introduce θ-dependent jumps and define the first-order condensation flux δD through saturation conditions that are not obviously compatible with a single m = 1 harmonic. The nonlinear residual λ1 in Eq. (C16) already contains wave-number-two structure (Figure 7), so the completeness of the m = 1 ansatz at first order should be justified explicitly, or the error from neglected longitude harmonics should be quantified for the first-order state.

minor comments (4)

[Section 2.3 heading] The heading contains a typo: 'Nondimensionlization' should be 'Nondimensionalization'.
[Table 1] The table header reads 'T able 1'; this should be corrected to 'Table 1'.
[Figure 7 caption] The caption uses 'phi' instead of the mathematical symbol φ; this makes the sentence hard to parse and should be fixed.
[Abstract and Section 3.3] The abstract states that the expansion is with respect to 1/Ro where Ro exceeds unity for typical lava planets, but the expansion parameter in Eq. (9) is defined with the sound speed, not the flow speed. The authors should clarify the relation between tilde_omega and the Rossby number actually used in the lava-planet literature, since the local ratio of Coriolis to advection terms is 2 tilde_omega / tilde_V^(0), which is not uniformly small.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the first-order asymmetric flow is derived from the stated conservation equations, with the axisymmetric state as an input rather than as the predicted quantity.

full rationale

The paper's central claim is that planetary rotation, through the Coriolis force, generates an O(tilde_omega) azimuthal circulation that breaks substellar-point symmetry. This is not true by construction: the governing equations (Eqs. 1-3) include the Coriolis term, and the perturbation expansion in Eq. 9 is substituted into those equations to obtain linearized first-order ODEs (Eq. 12) with forcing terms such as -2 tilde_V^(0) tilde_P^(0) sin^2 theta. The first-order variables (u1, V1, P1, T1) are then solved, not fitted. The sinusoidal longitude dependence in Eq. 11 is an ansatz, but it is the natural separable form forced by the cos phi/sin phi structure of the Coriolis term in a system whose coefficients are phi-independent, and the paper checks consistency via residuals in Appendix C, explicitly reporting where the truncation fails (theta >= 130 degrees). The zeroth-order axisymmetric solution from Kang et al. (2021) is an input to the perturbation, not the predicted result; although Kang et al. includes co-authors of this paper, it is an independently published base model, and the asymmetric component is not reduced to that citation. The exponential fits in Figure 5 are labeled as an empirical extrapolation and explicitly described as a hypothesis to be verified by future work, not as a derived prediction. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked. The concern that tilde_omega may not be sufficiently small for typical lava planets is a validity or correctness limitation, acknowledged by the authors in Section 3.3 and Appendices C-D, but it is not circularity. The derivation is self-contained relative to the stated assumptions, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the 1D axisymmetric base model of Kang et al. (2021), the small-tilde_omega expansion, and several physical simplifications (single-component, transparent, steady, hydrostatic atmosphere; prescribed surface temperature). The extrapolation to massive planets uses empirical fits to the model's own output. No new physical entities are introduced.

free parameters (4)

Achem, Bchem (chemical equilibrium vapor pressure parameters) = Achem = 10^9.6 Pa, Bchem = 38000 K
Taken from Schaefer & Fegley (2009) and Kang et al. (2021); fitted to bulk silicate vapor pressure data and used to compute Pchem(Ts), which controls the evaporation flux.
Asat, Bsat (saturation vapor pressure parameters) = Asat = 10^9.54 Pa, Bsat = 12070.4 K
Adopted from sodium atmosphere models (Castan & Menou 2011; Kang et al. 2021); fitted to saturation vapor pressure data and used for condensation.
Extrapolation fit coefficients Ai, Bi, Ci = not tabulated; fit to model output for M*/M_sun <= 0.25
Used in Section 3.3 to extrapolate first-order amplitudes to planets more massive than Earth; the fitted form is empirical and not derived from the governing equations.
Sticking coefficient alpha = 1
Chosen by hand for all simulations; alpha affects the mass exchange flux and the first-order amplitudes, and lower values are discussed but not used in the standard case.

assumptions (5)

domain assumption Steady-state, single-component, hydrostatic, radiatively transparent mineral-vapor atmosphere
Invoked in Section 2.1; excludes time dependence, vertical structure, radiative transfer, and multi-species effects.
domain assumption Surface temperature is set by local radiative balance F* + Fi = sigma*Ts^4, ignoring atmosphere-surface energy exchange
Section 2.1; this fixes Ts(theta) and therefore the vapor pressure field independently of atmospheric dynamics.
ad hoc to paper Variables expand in powers of small tilde_omega and are truncated at first order
Equation 9; the entire framework depends on tilde_omega being small, which is not guaranteed for all target planets.
ad hoc to paper First-order fields have the sinusoidal longitude form in Eq. 11
Section 2.3; a single-harmonic ansatz consistent with the Coriolis forcing but not derived from the nonlinear equations.
domain assumption Condensation flux D enters only the energy equation; first-order saturation condition approximated by dropping the second term in Eq. A2
Section 2.3 and Appendix A, following Kang et al. 2021; the dropped term is claimed to be small but is not quantified.

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Cite this review

Pith. "Pith review of Influence of Planetary Rotation on Supersonic Flow of Lava Planets: A Two-Dimensional Horizontal Model Analysis." pith.science (2026). https://pith.science/paper/GUNO2LFK

@misc{pith2026250104350,
  author       = {Pith},
  title        = {Pith review of: Influence of Planetary Rotation on Supersonic Flow of Lava Planets: A Two-Dimensional Horizontal Model Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUNO2LFK}},
  note         = {Machine review of arXiv:2501.04350}
}

read the original abstract

The study of lava planets has attracted significant attention recently because of their close proximity to their host stars, which enhances their detectability for atmospheric characterization. Previous studies showed that the atmospheric flow becomes supersonic if the atmosphere was dominated by rocky vapor evaporated from the magma ocean around the substellar point of small lava planets. These studies often assumed an axisymmetric flow about the axis from the substellar point to the antistellar point but ignored the effect of planetary rotation on the climate. The spin rate of lava planets can be rather fast due to their close-in orbits, which can break the aforementioned symmetry and induce the asymmetric flow component. Here, we introduce a two-dimensional framework to explore the influence of planetary rotation on the atmospheric dynamics of these lava planets for the first time, and assess the sensitivity and range of application of our model. Starting from the established one-dimensional axisymmetric atmospheric solution, we obtain the governing equation for the asymmetric flow by expanding with respect to 1/Ro (Ro denotes Rossby number and exceeds unity for typical lava planets). The asymmetric component of supersonic flow is pivotal for future research on the observation of these atmospheres, flow patterns of the magma ocean currents driven by atmospheric winds, and deformation of the planetary shape over long timescales.

Figures

Figures reproduced from arXiv: 2501.04350 by the authors.

**Figure 1.** Tidally locked coordinate (θ, ϕ) defined on lava planets in this work. The positive direction of transport flow V and circulation flow u are defined along the direction of θ and ϕ, respectively. The angular velocity of planetary rotation points along θ = π/2 and ϕ = 0. The host star is in the direction of θ = 0. We define the arc with ϕ = 0 and 0 < θ < π/2 as the prime meridian, then the hemisphere with sin ϕ ≤ 0 (i… view at source ↗

**Figure 2.** Upper panels: atmosphere fields as a function of tidally locked latitude θ for the zeroth-order state (left) and the first-order state (right) in the standard simulation, with dimension reinstated. Yellow curves represent the transport velocity V along eθ; blue curves represent the surface pressure P; light green curves represent the atmospheric temperature T; red curve represents the surface temperature Ts (left); … view at source ↗

**Figure 3.** Comparison of the first-order circulation velocity ωau1(θ) in the standard simulation between the exact solution (blue) and the approximation solution (red). The exact solution is the same as the green curve in the upper right panel of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Distribution of ∆1, ∆2, ∆3, ∆4 at θ = π/2 as a function of M∗ and Ts0, when M∗ varies from 0.01M⊕ and 0.2M⊕ and the substellar temperature Ts0 varies from 2000 K to 4000 K. The outliers of ∆4 when M∗ = 0.2M⊕, Ts0 = 2400 K, 2800 K, 3200 K, 3600 K are caused by the atmos…

**Figure 5.** Figure 5: ∆2(θ) = V1/V (0) , ∆3(θ) = −P1/P(0) , ∆4(θ) = −T1/T(0) as a function of M∗ when θ = π/3 (upper panels) and θ = π/2 (lower panels). The numerical solutions are represented by colored dots with various substellar surface temperatures (blue, orange, green, red, purple, an…

**Figure 6.** Figure 6: First-order solutions with various starting point θc for numerical integration in our standard simulation. The numerical solutions beyond θ > 10◦ are insensitive to the choice of θc. In each panel, the light blue, red, gray, yellow, dark blue, green color represent θc …

**Figure 7.** Figure 7: Surface distribution of the residual term λ1 in Eq. C13 (left panel), mass exchange flux between the surface and atmosphere F (middle panel), and λ1/F (right panel) in our standard simulation. λ1 is mainly made of wave-number-two component along the tidally locked long…

**Figure 8.** Figure 8: Surface distribution of the residual term λ22,2D in Eq. C14 (left panel), Coriolis term in the first-order momentum equation λ22,1D (middle panel), and their ratio λ22,2D/λ22,1D (right panel) in our standard simulation. λ22,2D is mainly made of wave-number-two componen…

**Figure 9.** Figure 9: Real part of the complex frequency σ as a function of the tidally locked latitude θ in our standard simulation, when m = 0 (left panel) and m = 1 (right panel). The four blue lines in each panel represent the four solutions of σ in Eq. D28. The zero value of σ is marke…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

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[2] [2]

write newline

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[3] [3]

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2021

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