Improving Our Knowledge of the Solar Near-Surface Shear Layer: The Special Case of the Leptocline

Alexander Kosovichev; Irina Kitiashvili; Jean-Pierre Rozelot

REVIEW 3 major objections 5 minor 2 cited by

Improving Our Knowledge of the Solar Near-Surface Shear Layer: The Special Case of the Leptocline

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The Sun's Near-Surface Shear Layer contains a distinct, about 8-Mm-deep substructure—the leptocline—where the rotational shear sharply intensifies and the strongest solar-cycle variations of the seismic radius occur.

desk verdict A candid conference summary of the leptocline case: useful new gradient maps, honest caveats, but the quantitative depth claims need a resolution check before they are taken as established. read the letter →

arxiv 2501.08021 v2 pith:3FL3QDFQ submitted 2025-01-14 astro-ph.SR

Jean-Pierre Rozelot , Alexander Kosovichev , Irina Kitiashvili This is my paper · ORCID

classification astro-ph.SR

keywords Near-SurfaceShearLayerleptoclinesolarrotationgradienthelioseismologyseismicradiuscycletorsionaloscillations

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 assembles global helioseismic observations from two decades (1996–2024) to argue that the Sun's Near-Surface Shear Layer (NSSL) is not a simple monotonic shear but contains a distinct, about 8-Mm-deep substructure just below the surface, which the authors call the leptocline. In this layer the radial gradient of rotation steepens sharply, enhanced turbulent convection from the hydrogen ionization zone modifies the stratification, and the strongest solar-cycle variations of the Sun's seismic radius occur. A sympathetic reader would care because the leptocline may be the place where the dynamo-generated magnetic field is concentrated and where sunspot-forming regions first feel the rotational shear, potentially linking the observed magnetic butterfly diagram to subsurface dynamics. The paper combines global helioseismology, local ring-diagram analyses, and 3D radiative hydrodynamics simulations to make the case, and notes that the true gradients may be even stronger than inversion-resolved values.

What carries the argument

The central object is the leptocline: a shallow, sharp rotational-shear layer about 8 Mm deep at the top of the solar convection zone (roughly $0.985$–$1.0\,R_\odot$), named by analogy with the tachocline. It is characterized by a steep radial gradient of rotation, enhanced turbulent convection rooted in the hydrogen and helium ionization zones, and self-organized meridional flows. The argument is carried by three complementary tools: (1) global helioseismic inversions of rotation splittings from SoHO/MDI and SDO/HMI data, which provide the average gradient structure and its time-latitude evolution; (2) f-mode frequency inversions, which track the solar-cycle displacement of subsurface layers and define the seismic radius; and (3) 3D radiative hydrodynamics simulations, which reproduce the leptocline and suggest its origin in anisotropic turbulent downdrafts that overshoot from the H i/He i ionization zone into the layer below. The simulations also indicate that the true rotational gradient in the leptocline may be as steep as about $-4$, much larger than the smoothed inversion values.

What would settle it

An observation that resolves the top 10 Mm of the Sun with high-degree modes (angular degree far beyond 300, as local helioseismology approaches) and finds that the logarithmic radial gradient of rotation does not steepen near the H i/He i ionization zone—or that the steepening is a fixed artifact of the inversion—would falsify the claim that the leptocline is a distinct shear layer. Concretely, if a resolved inversion shows $d\log\Omega/d\log r$ remaining near $-1$ continuously from 30 Mm to the surface, with no sharp excursion around 3–8 Mm, the leptocline as a physical layer would be ruled out.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Sun's Near-Surface Shear Layer has a distinct substructure, the leptocline, which occupies the top approximately 8 Mm of the convection zone (roughly between $0.985\,R_\odot$ and $1.0\,R_\odot$). In this layer the logarithmic radial gradient of the rotation rate, $d\log\Omega/d\log r$, increases sharply from about $-1$ in the deeper NSSL to about $-1.5$ at the equator and low latitudes, while at higher latitudes it becomes positive, so rotation speeds up toward the surface; the inversion results are smoothed by averaging kernels, so the actual gradients may be considerably larger. The leptocline coincides with the hydrogen and helium ionization zones, where enhanced anisotropic turbulent overshooting produces strong density fluctuations and turbulent mixing. The paper further reports that the rotational gradient in and just below the leptocline varies with the solar cycle: below the leptocline the enhanced gradient follows the magnetic butterfly diagram, whereas inside the leptocline the pattern is more complex and resembles the overlapping extended solar cycles of the torsional oscillations. Finally, inversion of f-mode frequencies shows the solar-cycle variations of the seismic radius are strongest at a depth of about 5 Mm, in the middle of the leptocline, with the Sun's average radius shrinking by a few kilometers near activity maxima.

Load-bearing premise

The results rest on helioseismic inversions whose averaging kernels smooth the sharpest near-surface structure, so the leptocline's reported gradients and their cycle variations could be diluted versions of stronger underlying values.

Editorial extensions

If this is right

Solar rotation models must treat the leptocline as a distinct layer: the sharp shear there means the NSSL is not a single power-law gradient but a two-step structure near the surface.
The enhancement of the rotational gradient in sunspot-forming regions, with a butterfly-like pattern of enhanced gradient below the leptocline, implies the leptocline participates in the magnetic activity cycle, likely in the formation or emergence of active regions.
The solar-cycle seismic-radius variations peak at about 5 Mm depth, inside the leptocline, meaning the Sun's small activity-related contractions are driven by subsurface stratification changes in this layer rather than by a global radial change.
Because inversion averaging kernels smooth sharp structures, the actual leptocline gradient may be significantly steeper than the reported $-1.5$, which strengthens the layer's dynamical importance for shear-driven instabilities and dynamo action.
The convergence of global helioseismology, local ring-diagram analysis, and 3D simulations on a shallow shear-enhanced layer suggests the leptocline is a physical feature, not an inversion artifact.

Reading between the lines

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

If the leptocline is a generic feature of cool stars with near-surface convection and hydrogen ionization zones, similar sharp shear layers should appear in other solar-like stars and could be probed through high-degree asteroseismic rotational splittings.
The anti-correlation between seismic radius and activity, strongest at 5 Mm, offers a testable energy-reservoir mechanism: if the leptocline stores or releases energy with the cycle, total irradiance variations and radius changes should track the layer's density and temperature perturbations in f-mode inversions.
The discrepancy between global-inversion and ring-diagram reversal latitudes of the gradient is a concrete place to look: a resolution-aware joint inversion could show whether the high-latitude positive gradient is real or a smoothing artifact, and whether it drifts with the extended cycle.
If the leptocline's gradient is as steep as simulations indicate, the magneto-rotational instability in this layer could be strong enough to matter for near-surface dynamo models, making the leptocline a boundary condition for theories that place dynamo action close to the surface.

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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

Short version: this is a proceedings-style synthesis of the leptocline argument, with one genuinely new piece—a 1996-2024 averaged map of the radial gradient of rotation from public SoHO/SDO inversions—plus time-latitude diagrams connecting the gradient to the butterfly diagram. What is new is largely descriptive. The conceptual load is carried by earlier papers (Kitiashvili et al. 2023; Komm 2022; Rabello Soares et al. 2024; Kosovichev & Rozelot 2018), and the paper is transparent about that.

Credit where due: the new averaged maps are a reasonable thing to have in the literature, the text explicitly warns that global inversions with l≤300 smooth the near-surface gradients and that true values may be larger, and the authors flag the mismatch between global and local helioseismic patterns rather than papering over it. The writing is clear and the citation pattern is fine.

The soft spots are real, and the main one is exactly what the stress test says. The paper spends Figures 2-4 presenting the leptocline's 8 Mm depth, its latitudinal gradient reversal, and its solar-cycle variations as if these were read directly off the inversions, despite the admission that the averaging kernels cannot resolve 8 Mm near the surface. If the kernel width there is comparable to the claimed layer thickness, the apparent features—especially the sign reversal at high latitude—could be inversion artifacts. A synthetic recovery test (invert a known leptocline-like profile with the same mode set and see what comes back) would settle this. Until then, the quantitative depth and amplitude claims are conditional. That is a load-bearing soft spot, but not a fatal one, because the local helioseismology and the 3D simulations independently point to a shallow sharp shear layer.

A second, smaller issue: Section 5 says the simulated gradient of about -4 is "in agreement with observations," but the global inversions show about -1.5 and the local ring-diagram values about -2.6. Those differ by a factor of 1.5-2.5. Agreement is too strong a word.

Who is this for? Readers who want a compact orientation to the leptocline discussion and a first look at the long-baseline gradient maps. It is not the definitive paper on the subject. If submitted as a regular journal article, it should be refereed: the central claim is important enough, and the resolution issue is serious enough, that the quantitative conclusions need to be checked before being cited as established. Recommendation: send it to review, and ask for the resolution test and a more careful simulation-observation comparison.

Referee Report

3 major / 5 minor

Summary. This proceedings paper analyzes global helioseismic rotation inversions from SoHO/MDI and SDO/HMI for 1996–2024, supplemented by local helioseismology and 3D radiative hydrodynamic simulations, to argue that the Near-Surface Shear Layer (NSSL) contains a distinct shallow substructure, the 'leptocline', approximately 8 Mm deep. The paper reports that the radial gradient of rotation, d log Ω / d log r, is strongly negative at low latitudes in the leptocline, reverses sign at high latitudes, varies with the solar cycle in sunspot-forming regions, and that f-mode-derived seismic radius variations are strongest near 5 Mm depth, inside the leptocline. The manuscript is explicitly built on prior work by the same authors and collaborators, including the naming of the leptocline and the simulations, and it is candid about the limited spatial resolution of global helioseismic inversions.

Significance. If the quantitative claims hold, the leptocline would be a dynamically important shallow layer controlling angular momentum transport, the formation of sunspots and active regions, and the solar-cycle variation of the seismic radius. The paper benefits from the convergence of global helioseismology, local ring-diagram measurements, and 3D simulations, and it explicitly cautions that inversion kernels smooth sharp gradients. However, the central quantitative content—the 8 Mm depth, the gradient amplitudes, and the latitude/time structure shown in Figs. 2–4—is not self-contained: it depends on low-resolution global inversions and on prior publications. The paper is a useful synthesis and a plausible qualitative case, but its specific quantitative characterization of the leptocline needs a resolution validation before it can be accepted as established by the new analysis.

major comments (3)

[§2, Figs. 2–3] The global helioseismic inversions use only modes with angular degree up to l=300, and the paper itself states that 'these data do not resolve sharp variations near the surface' and that gradients are smoothed by averaging kernels. This limitation is load-bearing because the central quantitative claims—the 8 Mm leptocline depth, the gradient value of about -1.5 at low latitudes, and the sign reversal at high latitudes—are read directly from these smoothed profiles. Please add a synthetic-recovery test: construct a rotation profile containing a sharp 8-Mm leptocline with known gradient values, synthesize rotational splittings for the same mode set and inversion procedure, and compare the recovered gradient profiles with the input. If the averaging-kernel width near r/R = 0.985–1.0 is comparable to or larger than 8 Mm, the apparent depth, amplitude, and latitudinal structure in Figs. 2–3 are not established by this data set alone. The qualitative existence of a shallow shear layer is independently supported by local helioseismology and simulations, but the quantitative characterization presented here requires this test.
[§3, Fig. 4] The time-latitude maps of d log Ω / d log r in the leptocline (Fig. 4c) are derived from the same low-resolution global inversions. The claim that the gradient is enhanced in the leptocline during activity minima as well as maxima, and the contrast between the 'below leptocline' (Fig. 4b) and 'in leptocline' (Fig. 4c) patterns, could be affected by changes in the radial averaging kernels with time or with the mode set. Please show that these patterns survive the synthetic-recovery test described above, or at least provide the radial averaging-kernel widths for the relevant depths and epochs. Without this, the apparent solar-cycle contrast may reflect a varying sensitivity rather than a physical change in the leptocline gradient.
[§4, Fig. 5] The inference that solar-cycle variations of the seismic radius are strongest at about 5 Mm depth, inside the leptocline, relies on f-mode frequency shifts, and the paper itself notes that magnetic field and temperature effects are difficult to separate. Since Fig. 5d is used to support the leptocline's role in cyclic changes, the authors should either quantify the likely systematic error from magnetic and thermal contamination (for example, using sensitivity kernels or comparing with p-mode-based radius estimates) or explicitly soften the claim to say that the variations are 'consistent with' a maximum inside the leptocline rather than 'strongest at' 5 Mm. As written, the quantitative depth localization is not supported by the analysis presented.

minor comments (5)

[§1] The word 'superadiabiatic' should be 'superadiabatic'.
[§3] The phrase 'near-shear shear layer' should be 'near-surface shear layer'.
[§6] The expansion 'Near Sub Surface Layer' should be 'Near-Surface Shear Layer' to match the acronym NSSL introduced in the abstract and used throughout the paper.
[References] Several references are incomplete, e.g., Kholikov and Hill (2008) lacks an article title, and Javaraiah (2003) lacks an article title; the reference list should be completed for consistency.
[Figure 2 and 3 captions] The captions and panel descriptions for Figures 2 and 3 appear somewhat scrambled: the text refers to a latitude dependence at three depths, but the Figure 2 caption only describes depth. Please ensure each figure panel is described by its own caption.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the central claims rest on independent helioseismic data and simulations, with only minor self-citation.

full rationale

I examined the derivation chain for reductions by construction. Section 2's radial-gradient maps are computed by standard inversions of public SoHO/MDI and SDO/HMI rotational splittings; no parameter is fitted to the leptocline depth or gradient and then re-predicted. The leptocline's ~8 Mm depth and strong gradient are attributed to independent ring-diagram analyses (Komm 2022; Rabello Soares et al. 2024) and to 3D RHD simulations (Kitiashvili et al. 2023); the cited simulation paper is by two of the present authors, but it is a separate, externally checkable calculation whose assumptions do not include the target result. Section 4's seismic-radius variations are adapted from Kosovichev & Rozelot (2018), a prior f-mode analysis, not from the same gradient data used to define the leptocline. The paper's own caveat that l<=300 global inversions cannot resolve sharp near-surface gradients is a resolution limitation, not a circularity: it weakens quantitative claims but does not make them equivalent to their inputs. Self-citations are frequent but none is load-bearing in the sense of forbidding alternatives or supplying the sole justification for a premise.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

No free parameters are fitted in this paper; it is an observational and simulation synthesis. The main assumptions are the resolution limits of helioseismic inversions, the interpretation of f-mode shifts as radius changes, and the extrapolation of a single local simulation to the Sun. The leptocline itself is an established concept from earlier work, not invented here.

assumptions (3)

domain assumption Helioseismic inversion of rotational frequency splittings yields the internal rotation rate with well-characterized averaging kernels.
Relied on throughout Section 2; the paper explicitly notes that the kernels smooth sharp near-surface gradients, limiting resolution.
domain assumption Variations in f-mode frequencies primarily reflect changes in the seismic radius and subsurface stratification.
Used in Section 4 to interpret f-mode inversions as radius changes; the paper admits that magnetic field and temperature effects are difficult to separate.
domain assumption A single local 3D radiative hydrodynamic simulation at 30 degrees latitude can represent the general leptocline structure of the Sun.
Section 5 generalizes the Kitiashvili et al. 2023 simulation to describe the leptocline's origin and properties without a full global simulation.

invented entities (1)

leptocline independent evidence
purpose: A shallow ~8 Mm substructure of the NSSL characterized by sharp rotational shear, enhanced turbulence, and ionization-zone overshooting, used to organize observed gradients and simulation results.
The leptocline is supported by global helioseismology, ring-diagram analyses, and 3D simulations, so it is not a pure postulate, though its physical origin and exact role remain hypotheses.

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Cite this review

Pith. "Pith review of Improving Our Knowledge of the Solar Near-Surface Shear Layer: The Special Case of the Leptocline." pith.science (2026). https://pith.science/paper/3FL3QDFQ

@misc{pith2026250108021,
  author       = {Pith},
  title        = {Pith review of: Improving Our Knowledge of the Solar Near-Surface Shear Layer: The Special Case of the Leptocline},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FL3QDFQ}},
  note         = {Machine review of arXiv:2501.08021}
}

read the original abstract

The discovery of the solar activity cycle was linked from the outset to the observation of the temporal variability of sunspots, which we know to be the result of complex processes associated with the dynamics of inner layers. Numerous recent studies have highlighted changes in the Sun's Near-Surface Shear Layer (NSSL), pointing to the role of the leptocline, a shallow and sharp rotational shear layer in the top around 8 Mm. The leptocline, mainly characterized by a strong radial rotational gradient at middle latitudes and self-organized meridional flows, is the cradle of numerous phenomena: opacity, superadiabaticity, and turbulent pressure changes; the hydrogen and helium ionization processes; a sharp decrease in the sound speed; and, probably, variations of the seismic radius associated with a nonmonotonic expansion of subsurface layers with depth. In addition, the leptocline may play a key role in forming the magnetic butterfly diagram. Such results are a starting point for further systematic investigations of the structure and dynamics of this layer, which will lead to a better understanding of solar activity.

Figures

Figures reproduced from arXiv: 2501.08021 by the authors.

**Figure 1.** The mean rotation rate, Ω, and the radial gradient, d log Ω/d log r, averaged over all SoHO/MDI and SDO/HMI measurements in 1996-2024, as a function of radius and latitude. a-b) cross-section views of the rotation rate and the gradient (the shaded region is where the inversion results are uncertain); c) the mean rotation rate at six latitudes indicated in the figure; d) the radial gradient as a function of radius at… view at source ↗

**Figure 2.** The radial gradient of the mean rotation rate, averaged over all SoHO/MDI and SDO/HMI measurements in 1996-2024, as a function of depth in the Near-Surface Shear Layer (NSSL) at three latitudes indicated in the figure. −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 dlog( W )/dlog( r ) Radial gradient of the solar rotation rate −60 −40 −20 0 20 40 60 latitude (deg) 0.95 0.96 0.97 0.98 0.99 1.00 r/R [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 3.** Variations of the rotational gradient, d log Ω/d log r, in the leptocline a) with the depth below the solar surface at three latitudes, b) with latitude at three depths (shown in the figure); c) with latitude-radius diagram of the rotational gradient in the NSSL (0.96-1.0 R⊙) and leptocline (0.99-1.0 R⊙). in more detail. The gradient that remains constant, ≃ −1, in the deep NSSL sharply increases its negative value … view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: a) The magnetic butterfly diagram for the radial component of magnetic field calculated from the SoHO and SDO line-of-sight magnetic field data, assuming that the magnetic field on the solar surface is predominately radial; b-c) the time-latitude diagrams for the rota…

**Figure 5.** Figure 5: Variations of the seismic radius of subsurface layers during Solar Cycles 23 and 24: a) the sunspot number of these cycles; b) the time-depth diagram of subsurface displacements, δr, inferred from f-mode frequencies obtained from the SoHO and SDO data; c) the variation…

**Figure 6.** Figure 6: Mean radial profiles of a) deviations of the azimuthal flow speed from the imposed rotation rate at 30 degrees latitude (red curve) and the adiabatic index, Γ1 (blue curve); b) the radial gradient of the rotation rate, defined as ∂ log Ω/∂ log r (red curve), and the RM…

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 2 Pith papers

[1]

diﬀerential rotation

Introduction In recent years, numerous studies have focused on the physical conditions prevailing in the Sun’s subsurface layers for at least two reasons. The ﬁrst a ddresses the problem of how both the physical conditions in subsurface layers of the Sun an d the nature of the magnetic ﬂux tubes of active regions are reﬂected in the structur e and behavio...

work page 1992
[2]

1995) and the Solar Dynamics Observatory (SDO; Scherrer et a l

Radial Gradients of Solar Rotation: T achocline and Leptocline The internal rotation of the Sun has been observed almost uninter ruptedly since 1996 from two space missions, the Solar and Heliospheric Observatory (S oHO; Scherrer et al. 1995) and the Solar Dynamics Observatory (SDO; Scherrer et a l. 2012) as well as from the ground-based Global Oscillatio...

work page 1995
[3]

torsional oscillations

Solar-Cycle V ariations of the Rotational Gradient Diﬀerential rotation also varies with time and typically reﬂects the so lar cycle. After subtracting the mean rotation, the residual component revealed alternating zones of fast and slow ﬂow bands, discovered by Howard and Labonte (1980) and called “torsional oscillations” because of their cyclic variatio...

work page 1980
[4]

V ariations of the Helioseismic Radius of the Sun With Respect to the Leptocline The solar-cycle variations of the Sun’s rotation rate and its gradien t in the NSSL and the leptocline are accompanied by structural changes related to t he dynamo-generated magnetic ﬁelds emerging on the solar surface. The subsurface mag netic ﬁeld has not been measured by h...

work page 2018
[5]

Radiative Hydrodynamics Simulations of the Leptocline Kitiashvili et al. (2023) analyzed realistic 3D radiative hydrodynamics simulations of solar subsurface dynamics in the presence of rotation in a local dom ain 80 Mm wide and 25 Mm deep, located at 30 degrees latitude. The simulations revealed the development of a shallow 8-Mm deep substructure of the ...

work page 2023
[6]

extended

Conclusions In summary, the results of global and local helioseismology as well as 3D radiative hydrodynamic simulations show that the NSSL (Near Sub Surface La yer) occupying the top 15% of the solar convection zone, the depth range ≈ 30 − 35 Mm) has a distinct substructure, the leptocline, which is about 8 Mm deep and charact erized by enhanced turbulen...

work page doi:10.48550/arxiv.astro-ph/970 1998

Pith tools

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

[1] [1]

diﬀerential rotation

Introduction In recent years, numerous studies have focused on the physical conditions prevailing in the Sun’s subsurface layers for at least two reasons. The ﬁrst a ddresses the problem of how both the physical conditions in subsurface layers of the Sun an d the nature of the magnetic ﬂux tubes of active regions are reﬂected in the structur e and behavio...

work page 1992

[2] [2]

1995) and the Solar Dynamics Observatory (SDO; Scherrer et a l

Radial Gradients of Solar Rotation: T achocline and Leptocline The internal rotation of the Sun has been observed almost uninter ruptedly since 1996 from two space missions, the Solar and Heliospheric Observatory (S oHO; Scherrer et al. 1995) and the Solar Dynamics Observatory (SDO; Scherrer et a l. 2012) as well as from the ground-based Global Oscillatio...

work page 1995

[3] [3]

torsional oscillations

Solar-Cycle V ariations of the Rotational Gradient Diﬀerential rotation also varies with time and typically reﬂects the so lar cycle. After subtracting the mean rotation, the residual component revealed alternating zones of fast and slow ﬂow bands, discovered by Howard and Labonte (1980) and called “torsional oscillations” because of their cyclic variatio...

work page 1980

[4] [4]

V ariations of the Helioseismic Radius of the Sun With Respect to the Leptocline The solar-cycle variations of the Sun’s rotation rate and its gradien t in the NSSL and the leptocline are accompanied by structural changes related to t he dynamo-generated magnetic ﬁelds emerging on the solar surface. The subsurface mag netic ﬁeld has not been measured by h...

work page 2018

[5] [5]

Radiative Hydrodynamics Simulations of the Leptocline Kitiashvili et al. (2023) analyzed realistic 3D radiative hydrodynamics simulations of solar subsurface dynamics in the presence of rotation in a local dom ain 80 Mm wide and 25 Mm deep, located at 30 degrees latitude. The simulations revealed the development of a shallow 8-Mm deep substructure of the ...

work page 2023

[6] [6]

extended

Conclusions In summary, the results of global and local helioseismology as well as 3D radiative hydrodynamic simulations show that the NSSL (Near Sub Surface La yer) occupying the top 15% of the solar convection zone, the depth range ≈ 30 − 35 Mm) has a distinct substructure, the leptocline, which is about 8 Mm deep and charact erized by enhanced turbulen...

work page doi:10.48550/arxiv.astro-ph/970 1998