Solar differential rotation driven by baroclinic forcing

G. Guerrero; J. Zhang; L. S. Menicucci; M. Dikpati; P. K. Smolarkiewicz; R. Hester

REVIEW 3 major objections 8 minor 299 references

Solar differential rotation driven by baroclinic forcing

T0 review · 3 major / 8 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The Sun's differential rotation—fast equator, slow poles—can be sustained by a small latitudinal entropy gradient in thermal wind balance, without needing vigorous convection or Reynolds stresses. The authors demonstrate this through 20 glo

desk verdict A careful spherical-shell demonstration that a prescribed baroclinic forcing can sustain solar-like differential rotation without strong convection, but the solar scenario rests on an unmodeled entropy gradient. read the letter →

arxiv 2607.19730 v1 pith:PEASFJEO submitted 2026-07-22 astro-ph.SR

L. S. Menicucci , G. Guerrero , M. Dikpati , R. Hester , J. Zhang , P. K. Smolarkiewicz This is my paper · ORCID

classification astro-ph.SR

keywords solardifferentialrotationbaroclinicforcingthermalwindbalanceconvectiveconundrumlatitudinalentropygradientglobalhydrodynamicsimulationsgyroscopicpumpingconvectionzone

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the Sun's observed differential rotation (equator rotating faster than poles, with radially aligned mid-latitude contours) can be generated and maintained without requiring strong turbulent convection and its associated Reynolds stresses. Instead, the authors propose that a background latitudinal entropy gradient, held in thermal wind balance, is enough to drive the flow. They test this idea with a suite of global hydrodynamic simulations spanning stable, adiabatic, and weakly unstable stratifications, finding that all produce a solar-like rotation profile. The result offers a way around the 'convective conundrum', where mixing-length theory predicts much stronger convection than helioseismology observes.

What carries the argument

The central object is the prescribed latitudinal potential-temperature gradient Θ_B, built by integrating the thermal wind balance relation (Eq. A1) against the helioseismic rotation profile. The stratification is enforced through a Newtonian cooling term in the potential-temperature equation (Eq. 3) with a long timescale (2.4–4.9 yr). The dynamical balance is expressed in the thermal wind equation: the centrifugal term C = r sinθ ∂Ω²/∂z is balanced against the baroclinic term B = (g/(Θ₀ r)) ∂Θ′/∂θ. The small residual C−B drives gyroscopic pumping, producing meridional circulation cells that redistribute angular momentum and sustain the differential rotation without the need for turbulent st

What would settle it

A direct helioseismic measurement of the latitudinal entropy gradient in the convection zone, showing it to be absent or far smaller than the thermal-wind value needed to balance the observed rotation, would falsify the mechanism. Alternatively, detection of Reynolds stress torques dominating over meridional-circulation torques in the Sun (as deduced from inversions of flow correlations) would contradict the paper's central claim.

Watch

Extended reading notes

Core claim

The central claim is that solar-like differential rotation can be generated and maintained in the convection zone without relying on Reynolds stresses from vigorous turbulent convection, provided a large-scale latitudinal entropy gradient exists and is sustained in thermal wind balance. In a rotating spherical shell with anelastic equations and a nudged potential-temperature profile, all simulations—whether stably stratified, adiabatic, or weakly superadiabatic—produce a rotation profile with a faster equator and slower poles. The torque analysis shows that the meridional circulation, rather than Reynolds stresses, is the main carrier of angular momentum in these runs; the Reynolds stresses

Load-bearing premise

The Sun's convection zone must actually maintain a persistent large-scale latitudinal entropy gradient, in thermal wind balance with the observed rotation, over timescales of years; if convection mixes that gradient away faster than it can be replenished, the proposed mechanism will not operate in the real Sun.

Editorial extensions

If this is right

If correct, the solar convection zone can be much less turbulent than mixing-length theory predicts, easing the convective conundrum.
The observed differential rotation profile becomes a signature of large-scale thermodynamic balance rather than a direct measure of convective Reynolds stresses.
The same baroclinic mechanism could operate in other solar-type stars, giving a common explanation for their rotation profiles.
Models of the solar dynamo, which rely on differential rotation to stretch magnetic fields, may need to account for a rotation profile that is thermodynamically constrained rather than convectively driven.
The work suggests that the radial tilt of the rotation contours at mid-latitudes emerges naturally from gyroscopic pumping of the baroclinic imbalance, offering a direct explanation for this long-puzzling feature.

Reading between the lines

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

A testable corollary of the paper's logic is that the observed solar rotation profile implies the existence of a sustained latitudinal entropy gradient of roughly the magnitude of Θ_B plotted in Fig. 2; if helioseismic or other measurements ever rule out such a gradient over multi-year timescales, the mechanism would be in conflict.
The paper's preference for weak or stable stratifications suggests that the Sun's interior may contain subadiabatic layers (as in the Deardorff-layer picture); an inference is that the location and sharpness of such layers would control the shape of the rotation profile and could be constrained by the observed radial behavior near the tachocline.
Extending this hydrodynamical framework to include magnetic fields, as the authors suggest, would test whether the baroclinically maintained rotation can survive Lorentz forces; one might predict that the magnetically active cycle modifies the entropy gradient and hence the rotation profile, providing a link to the observed torsional oscillations.
The paper does not prescribe the origin of the entropy gradient; an inference is that any convective or tachocline-driven process that creates a large-scale latitudinal entropy variation (e.g., rotational modulation of heat transport or entropy rain) would suffice, meaning the mechanism is robust to the exact source as long as the gradient persists.

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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

Honest, well-scoped numerical experiment. The new thing is not the thermal-wind idea — Balbus, Gunderson, Hester and others have been there — but the nonlinear spherical-shell demonstration. Across 20 runs spanning stable, adiabatic, and weakly unstable stratifications, the authors show that a prescribed latitudinal entropy gradient in thermal wind balance drives a steady, fast-equator/slower-pole rotation profile with mid-latitude radial contours, and their torque decomposition attributes the maintenance to meridional circulation rather than Reynolds stresses. That is real evidence for dynamical viability, and the diagnostics are careful: 456 years of evolution, 76-year averages, explicit equations and boundary conditions.

The soft spot is the load-bearing premise. The entropy gradient is not modeled; it is imposed as Θ_B, and that target is constructed from the very helioseismic Ω the paper aims to explain (Appendix A). The Newtonian cooling nudges toward it, so a solar-like outcome is partly inherited. The authors are open about this — they call it a controlled forcing and defer the origin — but it means the paper demonstrates consistency, not that the Sun possesses such a gradient. If convection mixes the gradient away, the mechanism fails. Their own results also show fragility: only weakly convective cases reproduce the observed radial increase at low latitudes, and strongly superadiabatic cases collapse to cylindrical contours. And the headline claim about not relying on Reynolds stresses is qualified even within the model, since §3 says Reynolds stresses help maintain that radial increase in the weakly convective cases.

Minor but real: no error bars on the time averages, no quantitative match metric against helioseismic Ω, and no released code or data. All addressable.

My overall take agrees with the reader's conditional verdict: this is a useful proof-of-concept that the baroclinic scenario is dynamically plausible, not a demonstration that it operates in the Sun. The paper is honest, internally consistent, and clearly scoped. It deserves a serious referee; the authors have done the work to make a referee's job straightforward. I'd want the entropy-gradient origin acknowledged more forcefully as a limitation, but it already is. Worth engaging with.

Referee Report

3 major / 8 minor

Summary. The paper investigates whether solar differential rotation can be sustained by a prescribed large-scale latitudinal entropy gradient rather than by Reynolds stresses from vigorous convection. Global anelastic EULAG-MHD simulations of a spherical shell (0.72–0.96 R_sun) are run for 20 combinations of polytropic index (stable, adiabatic, weakly unstable) and two Newtonian-cooling timescales, with a target potential-temperature profile Θ_B derived from helioseismic differential rotation via thermal-wind balance. Time-averaged results show a fast equator and slower poles across most runs, with radially aligned mid-latitude contours; torque analysis attributes the maintenance primarily to meridional circulation, with Reynolds stresses playing a smaller or auxiliary role. The authors conclude that baroclinic forcing can maintain solar-like differential rotation without strong convective Reynolds stresses, offering a possible resolution to the convective conundrum, while acknowledging that the physical origin of the imposed entropy gradient is not modeled.

Significance. The paper has genuine strengths: the equations, boundary conditions, grid, and parameter choices are fully specified; the parameter sweep is broad; and the simulations are evolved long enough for robust temporal averages. The torque decomposition and thermal-wind diagnostics are appropriate and clearly presented. If the claims are scoped carefully, the study is a useful dynamical proof-of-concept: it shows that a baroclinic state designed to be in thermal-wind balance is nonlinearly compatible with a fast-equator rotation profile across a range of stratifications. However, the central solar-application claim is weakened by the construction of Θ_B from the very differential-rotation profile the paper explains, and by the fact that the entropy gradient is externally maintained rather than shown to exist or persist in the Sun. The paper is therefore more convincing as a controlled dynamics study than as an explanation of the origin of solar differential rotation.

major comments (3)

[Appendix A, Eq. (A1); §4] The target profile Θ_B is constructed by integrating the thermal-wind balance using the observed helioseismic Ω profile (Eqs. A2–A3), and Eq. (3) relaxes the simulation toward this target. The solar-like Ω in the runs is therefore at least partly imposed: the forcing encodes the very observable being explained. The paper states explicitly that the physical origin of the entropy gradient is not modeled (§1, §4), but the abstract and conclusion present the result as an 'alternative scenario' for the solar rotation profile. As it stands, this is a demonstration of dynamical consistency, not a demonstration that the Sun possesses a sustained entropy gradient of the required amplitude and structure. Please narrow the claims accordingly, or provide a quantitative estimate of the required Θ_B and a discussion of whether such a gradient is consistent with observational constraints and with the r
[§2, boundary conditions after Eq. (3)] Eq. (3) relaxes Θ′ toward Θ_B, but the boundary conditions immediately below require Θ′=0 at r=0.72 and 0.96 R_sun. The target Θ_B shown in Fig. 2 has latitudinal structure at these radii (its latitudinal average is zero, but pointwise values are not), so the homogeneous Dirichlet condition is incompatible with the nudging target at the boundaries. This will create boundary layers in Θ′ and corresponding baroclinic torques. Please clarify how the boundary condition is imposed in the code, whether the source term is suppressed at the boundary cells, and whether the reported time averages—including Fig. 3 and the torque integrals of Fig. 4—are sensitive to this treatment.
[§3, first paragraph; §4] The paper's headline claim that solar-like differential rotation is maintained 'without relying on Reynolds stresses' is internally qualified. §3 states that 'weakly convective models ... are the only ones capable of reproducing the observed radial increase of Ω at low latitudes' and that 'the Reynolds stresses are responsible for torques capable of maintaining the radial increase of the rotation rate.' The radial increase at low latitudes is part of the observed solar profile, so the abstract and conclusions overstate the result. Please recast the claim to distinguish the baroclinically driven fast-equator/pole contrast from the low-latitude radial shear, which in the successful simulations is maintained with assistance from weak convective Reynolds stresses.

minor comments (8)

[Eq. (10)] The symbols C and B are used before they are defined; define them explicitly or insert a sentence introducing them.
[§2, paragraph on nudging] The phrase 'wavelength agnostic diffusivity' is vague. Since the Fourier-space form −τ^{-1}Θ̂′ is given, please explain the intended physical interpretation more concretely.
[Throughout] Spelling is inconsistent: 'superabadicity' vs. 'superadiabaticity' and 'heliosseismic' vs. 'helioseismic'. Please check the whole text.
[§4] The concluding paragraph ends with a stray 'a' after '...viable alternative to convection-dominated paradigms and contributes toward resolving long-standing discrepancies.'
[Fig. 2 caption] Please state the units and color-scale range for the potential-temperature panel; currently the reader cannot estimate the amplitude of Θ_B.
[Table 1] Entries in the second column like '1.5−1×10−4' are easy to misread as 1.5×10−4. Use explicit arithmetic (e.g., '1.4999') or add a space.
[References] In the reference for Stefan et al. 2026, 'Guerreo' appears to be a typo for 'Guerrero'. Please check.
[§2, numerical model] No explicit viscosity or diffusivity appears in Eqs. (1)–(3). If the runs rely on EULAG's numerical dissipation as an implicit subgrid model, please state this explicitly, since it is relevant for interpreting the Reynolds-stress torque decomposition.

Circularity Check

1 steps flagged · score 6.0 of 10

Solar-like rotation is partly an input: the prescribed entropy gradient Θ_B is derived from the observed helioseismic Ω profile, and the simulations are nudged toward it.

self definitional [Appendix A, Eq. (A1); Section 2, Eq. (3) and surrounding text]
"To obtain an entropy distribution compatible with the differential rotation profile inferred by helioseismic measurements, we integrate the thermal wind balance relation ... ΘB = C(r) − Θ0/g ∫ ∂Ω^2/∂z r dθ (A1). The nudging target profile Θ B implements an axis-symmetric potential temperature distribution in thermal wind balance with a velocity profile ... obtained by fitting the helioseismic observations ... into u 0 = a 0(r) + a 2(r) cos2(θ) + a 4(r) cos4(θ)."

Θ_B is constructed by integrating the thermal-wind balance from the observed helioseismic Ω profile (Eqs. A1–A3). The prognostic equation (3) then nudges Θ′ toward Θ_B on a τ ≈ 2.4–4.9 yr timescale. The simulated 'solar-like' rotation (fast equator, slow poles, radial contours) is therefore at least partly an input rather than an output: the forcing already encodes the target. The paper's own statement that the gradient is 'prescribed as a controlled forcing to test its dynamical consequences rather than to model its physical origin' confirms that the central driver is imported, not derived. Partial non-circular content remains in the departure from perfect TWB and the torque analysis.

full rationale

The paper makes a clear and honest distinction: it does not claim to derive the latitudinal entropy gradient from first principles, but rather prescribes it to test its dynamical consequences. However, the central result—that solar-like differential rotation can be generated without strong Reynolds stresses—is partially circular because the prescribed gradient itself is obtained by integrating the thermal-wind balance from the very helioseismic rotation profile that the simulations then reproduce. The Newtonian cooling term (Eq. 3) continuously relaxes the entropy perturbation toward this observed-derived target, so the pole-to-equator increase in Ω and the mid-latitude contour alignment are substantially built into the forcing. The non-vacuous content is the nonlinear dynamical consistency: the simulations show how the steady state departs from the imposed thermal-wind state via meridional circulation and how torques balance, which is a genuine result. There is no load-bearing self-citation or imported uniqueness theorem; the cited prior work (e.g., Hester et al. 2025) is used for comparison, not as a logical premise. The score of 6 reflects that one central 'prediction' (solar-like Ω) reduces by construction to the fitted input Θ_B, while the paper's own limitations statements are explicit and the dynamical analysis retains independent content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claim rests on: (1) a forcing profile Θ_B constructed from the observed rotation through the thermal-wind integral — the target embeds the helioseismic Ω fit; (2) a Newtonian-cooling device (Eq. 3) that maintains this baroclinicity on multi-year timescales, standing in for an unmodeled physical process; (3) standard anelastic/polytropic modeling choices. Roughly a dozen hand-set or fitted numbers (Δ∇ scan values, τ values, the six DR-fit coefficients, and the gauge C(r)) enter the setup. No new physical entity beyond the prescribed entropy gradient is introduced, and its existence in the Sun is unverified.

free parameters (4)

Helioseismic DR fit coefficients α0, α2, α4, β0, β2, β4 (Eq. A2/A3) = Not tabulated (multiple linear regression to Larson & Schou 2018 data)
These coefficients define the smooth rotation profile that, via Eq. (A1), generates the forcing Θ_B. The solar-like character of the target is thus baked in from the fit.
Gauge function C(r) of the thermal-wind integral = Chosen by imposing vanishing latitudinal average of Θ_B
Eq. (A1) determines Θ_B only up to an arbitrary radial function; the chosen gauge sets the radial structure of the forced baroclinicity and therefore shapes the simulated radial Ω profile.
Nudging timescale τΘ = 7.8×10^7 s (~2.4 yr) and 1.5×10^8 s (~4.9 yr)
Hand-chosen values controlling the stiffness of the thermodynamic constraint in Eq. (3). The paper shows results depend on this choice (strongly convective models change behavior with τ).
Ambient superadiabaticity Δ∇ (10 values via polytropic index m_a) = Values in Table 1, ranging from −1.5×10^−7 to −1×10^−4, plus 0
Exploration parameters that set whether the layer is convectively unstable, neutral, or stable. The solar-likeness of the outcome is sensitive to this scan: strongly unstable cases become cylindrical.

assumptions (6)

domain assumption Anelastic (Lipps–Hemler) approximation is valid for the subsonic solar convection zone flow.
Invoked in §2, Eqs. (1)–(3). Standard for global solar convection modeling but filters acoustic modes and assumes small perturbations around a reference state.
ad hoc to paper Newtonian cooling with τ ≈ 2.4–4.9 yr (Eq. 3) is a faithful stand-in for whatever physical process would hold the entropy gradient in place.
The relaxation term is a numerical device whose solar counterpart is not identified. Without it, the prescribed baroclinicity would diffuse away; the mechanism's persistence in the Sun is therefore assumed, not shown.
standard math The constructed state satisfies steady thermal-wind/geostrophic balance, so Θ_B from Eq. (A1) is the correct entropy pattern for the fitted Ω.
Appendix A integrates the thermal-wind relation B = C, a standard derivation under the stated assumptions of a purely azimuthal flow and steady balance.
domain assumption The background stratification is polytropic with base-state index m_0 = 1.5 (adiabatic), matched to the solar standard model at r_min.
Used in §2, Eqs. (5)–(6) and Fig. 1. The quasi-adiabatic solar convection zone motivates this choice; departures near the boundaries are neglected.
domain assumption Stress-free, impermeable boundaries with vanishing potential-temperature fluctuation at r_min and r_max.
Stated in §2. These boundary conditions affect the torque balance and the meridional flow structure in the simulated shell.
domain assumption Gravity follows an inverse-square law with mass enclosed at r_min (Eq. 4).
Stated in §2. Standard for a thin spherical shell in the solar interior.

invented entities (1)

Externally prescribed, sustained latitudinal entropy (potential-temperature) gradient Θ_B in thermal wind balance with the observed rotation
purpose: Provides the baroclinic forcing that generates and maintains solar-like differential rotation without convective Reynolds stresses
Θ_B is constructed from the observed Ω via Eqs. (A1)–(A3) and maintained by the nudging term of Eq. (3). No observation, mechanism, or falsifiable handle outside the model establishes that such a gradient exists in the Sun; the paper explicitly states its origin is left open (§4).

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Cite this review

Pith. "Pith review of Solar differential rotation driven by baroclinic forcing." pith.science (2026). https://pith.science/paper/PEASFJEO

@misc{pith2026260719730,
  author       = {Pith},
  title        = {Pith review of: Solar differential rotation driven by baroclinic forcing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEASFJEO}},
  note         = {Machine review of arXiv:2607.19730}
}

read the original abstract

A combination of recent observations and numerical simulations has called into question whether Sun's interior is characterized by strong turbulent convection, a problem known as convective conundrum. In light of a possible absence of vigorous convective motions, we examine whether the Sun's differential rotation, previously assumed to be tightly coupled to Reynolds stresses, may instead be sustained by the presence of a background latitudinal entropy gradient in thermal wind balance. By performing global hydrodynamical simulations of a rotating spherical shell representing the bulk of the solar convection zone, we demonstrate that solar-like differential rotation can be generated under a variety of thermal stratifications. This work proposes an alternative scenario for the origin of solar differential rotation that accommodates the previously reported discrepancies.

Figures

Figures reproduced from arXiv: 2607.19730 by the authors.

**Figure 1.** Top panel: polytropic background density, ρ0 (black line) and temperature, T0 (red line), profiles plotted as a solid line. The corresponding quantities extracted from the solar standard model of J. Christensen-Dalsgaard et al. (1996) are presented with dashed lines. Bottom panel: radial profiles of the ambient potential temperature for each class of simulations. The S, U and AD labels refer, respectively, to stab… view at source ↗

**Figure 2.** Angular velocities fitted from the heliosseismic data (T. P. Larson & J. Schou 2018) into Eqs. (A2) and (A3) (left), color limits are fixed to match the heliosseismic data. Potential temperature profile obtained after integration of Eq. (A1), using the fitted velocities, and imposition of a vanishing latitudinal average (right). The nudging target profile ΘB implements an axissymmetric potential temperature distri… view at source ↗

**Figure 3.** Time and longitudinal averages of the rotation rate (a, d, g), latitudinal velocity (b, e, h) and potential temperature perturbation (c, f, i) fields for three characteristic simulations chosen as so to represent the total set of simulations. Color levels on the first column is adjusted to match the limits of heliosseismic data (T. P. Larson & J. Schou 2018). Each row corresponds to a different simulation named aft… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Time averaged radial (top row) and latitudinal (bottom row) integrals of the Reynolds stresses defined in Eq. (9) for the same simulations of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Time and longitudinal averages of the centrifugal (panels (a), (d), (g)) and baroclinic (panels (b), (e), (h)) terms, defined in Eq. (10), as well as their difference (panels (c), (f), (i)) for the same set simulations of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Time and longitudinal averages of the rotation rate, latitudinal velocity and potential temperature perturbation for the models listed in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Residual. Physical Review , author =. 1967 , pages =. doi:10.1103/PhysRev.162.162 , language =

doi:10.1103/physrev.162.162 1967