Turbulent Elasticity of the Solar Convective Zone and the Taylor Number Puzzle

Peter Todd Williams

REVIEW

Turbulent Elasticity of the Solar Convective Zone and the Taylor Number Puzzle

Peter Todd Williams

Astrophysics · Submitted 23 Feb 2006 · 17 references

Pith's verdict: A clearly-written, honest research proposal that identifies a novel mechanism for the Taylor number puzzle, but the required stress magnitude is off by a factor of 10–30 (and likely more) and the paper doesn't close that gap.

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arXiv IDastro-ph/0602502 (version 1)

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CitationPeter Todd Williams. "Turbulent Elasticity of the Solar Convective Zone and the Taylor Number Puzzle." arXiv:astro-ph/0602502 (2006). https://pith.science/paper/K5I6YLYX

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Claim paper

Why Pith says this

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Editor's note

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Colleague,

Short answer: this is an honest, clearly written conference-style paper that proposes a genuinely new mechanism for the Taylor number puzzle, but the quantitative case doesn't close. The new bit is applying the viscoelastic Maxwell-stress model — from Williams' own earlier work and from Ogilvie and Longcope — to the SCZ's meridional circulation, and estimating the stress needed to cancel centrifugal driving. The paper frames the puzzle well, freely cites competing solutions (baroclinicity, AKA effect), and is explicit that the conclusion is a hypothesis. That is the good part.

The soft spot is the magnitude. The required streamwise elastic stress is νsγ², and the ratio to the viscous stress is sγ ≃ 10². The only quantitative anchor in the paper, MRI shearing-box simulations, gives streamwise/viscous stress ratios of 10^0.5–10^1 — a factor of 10–30 below what is needed. The author calls that 'encouraging,' but when the mechanism's entire purpose is to balance a known source term, a factor of 10–30 is the whole ballgame. Worse, a direct estimate of sγ in the SCZ with a convective turnover time or Alfvén crossing time gives sγ ~ 0.1–1, two to three orders below the requirement. The Alfvén-velocity estimate (v_A ~ few×10^4 cm/s) also gives a stress ratio (v_A/v_t)² ~ 10, not 100. So even if the viscoelastic analogy holds, the counter-circulation looks far too weak. The sign argument is plausible—tension stress decreasing with z gives the right positive vorticity—but the magnitude balance isn't closed.

The central mechanism leans on self-cited prior work for the stress closure. That isn't automatically a flaw, but the application here requires an extrapolation to sγ~100 that is not defended. The paper is really a research proposal, not a result.

Who would get value from it: people working on solar mean-field theory might find the hypothesis worth a thought, and it is a reasonable conference proceedings. I would not cite it as evidence for the mechanism, though I might cite it as a pointer if I discussed the puzzle.

If I were an editor, I'd accept it for a conference proceedings but not for a full refereed journal — the central quantitative claim is not established. That said, it deserves a careful reading by a sympathetic referee who can point the author toward what would be needed to make the case: a derivation or simulation that actually produces sγ~100.

Referee report

Significance

If the mechanism were quantitatively established, it would add a new physical ingredient—turbulent Maxwell stress with viscoelastic structure—to mean-field models of the SCZ and could plausibly help reconcile modeled and helioseismically observed rotation profiles without resorting to artificially low Taylor numbers. The paper is useful in clearly identifying a specific, falsifiable mechanism and in connecting the solar problem to laboratory viscoelastic secondary-flow reversal. However, the quantitative basis is presently weak: the key stress-ratio estimate is not derived, the only numerical comparison cited (MRI simulations) lies one to one-and-a-half orders of magnitude below the required value, and the underlying constitutive assumption is imported from prior work rather than independently justified in the SCZ context. The strength of the paper is its explicit hypothesis and its candid acknowledgment of the current order-of-magnitude gap; its weakness is that this gap is precisely the load-bearing quantity for the proposed solution to the Taylor number puzzle.

Major objections (4)

[Eq. (5) and surrounding paragraph] The central parameter sγ is asserted to be approximately 10², but this value is not derived from the governing equations; it is stated as the 'required' ratio. The only numerical estimate cited in the paper, from MRI shearing-box simulations, gives streamwise-to-viscous stress ratios of 10^0.5 to 10^1, a factor of 10 to 30 smaller than the required value. For a mechanism whose entire purpose is to balance a known centrifugal source, a factor of 10–30 in the balancing stress is the load-bearing question, and the paper does not resolve it.
[Viscoelastic closure and Eq. (5)] The constitutive assumption that the turbulent Maxwell stress contains a streamwise term proportional to the second Rivlin-Eriksen tensor is imported from Williams (2004, 2005) without derivation or independent justification in this paper. The polymer analogy is qualitative and does not establish that SCZ turbulence behaves as a viscoelastic fluid with the required Weissenberg number. Indeed, sγ ≈ 10² combined with γ ≈ 10^-6.5 s⁻¹ implies a relaxation time s ≈ 10^8.5 s, which is orders of magnitude longer than the convective turnover time of about 10^6 s; no physical mechanism is offered to account for such a long relaxation time.
[Sign argument around Eq. (4)] The direction of the effect relies on the claim that the magnitude of the shear, and hence the elastic stress a, decreases with height z in the northern mid-latitudes, so that ∂z(a/R) is negative and Eq. (4) gives a positive source. This is stated only as a 'tendency' and is not quantified with helioseismic data or a model profile. In addition, the text equates positive vorticity with a clockwise circulation, which is opposite to the standard convention in which positive ω_z corresponds to counterclockwise flow; as written, it is unclear whether the proposed stress counteracts or reinforces the centrifugal circulation.
[Alfvén-velocity estimate, same section as Eq. (5)] The two magnitude estimates in the paper are internally inconsistent. The Alfvén-velocity estimate gives v_A ≈ a few ×10^4 cm/s, which is only a few times the typical convective velocity v_t ≈ 10^4 cm/s in the SCZ, so the implied ratio (v_A/v_t)² is about 10, not 100. The sγ ≈ 10² estimate, by contrast, would require a magnetic field approximately an order of magnitude above equipartition for the quoted values of density and velocity. The statement that the resulting field of 10^1.5 kG 'does not seem unreasonable' is not supported by the quoted convection-zone parameters.

Minor objections (5)

[General] The paper is written in a deliberately informal conference style; phrases such as 'actually, that is a bit of a lie' are acceptable in a talk but should be removed for a formal journal article.
[Eq. (2)] The vorticity equation is written with an approximate equality because the viscous term is simplified; this is fine, but the notation should make clear that the same simplification is used in the subsequent discussion of the stress divergence.
[Definition of Maxwell stress] The definition 4π M_ij = B'_i B'_j does not specify whether M_ij is the trace-free part or includes the trace; the text later refers to isotropic stresses being ignored, but the tensor algebra should be stated explicitly to avoid ambiguity.
[Helioseismic support] The claim that ∂z|γ| < 0 in northern mid-latitudes should be supported by a citation to a specific helioseismic inversion or by a figure showing the rotation profile, since this sign is critical to the proposed mechanism.
[Taylor number definition] Equation (1) defines Ta using ν, but the text later uses the turbulent viscosity for the solar estimate; a sentence clarifying that ν is the turbulent viscosity in the SCZ estimate would prevent confusion.

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The paper

The paper claims

Turbulent magnetic fields, modeled as a viscoelastic fluid, may explain the Sun's rotation profile without an artificially small Taylor number.

The reading

In this preliminary study, the paper takes on the Taylor number puzzle: mean-field models of the solar convection zone reproduce the helioseismically observed rotation profile only if the Taylor number is made artificially small, whereas the realistic value $Ta\sim 10^6$ drives a meridional circulation so strong that it would destroy the differential rotation. The proposed missing ingredient is the turbulent Maxwell stress of the small-scale, tangled magnetic field, which the paper treats as a viscoelastic fluid rather than a purely viscous one. Under steady shear this stress develops a streamwise elastic tension proportional to the second Rivlin-Eriksen tensor, and the curl of its divergence sources meridional vorticity opposite in sign to the centrifugal driving. If that source is real, the calculated angular velocity profile should move toward the observed one without lowering $Ta$, and the required field strengths, of order $10^{1.5}$ kG at mid-convection-zone depths, are plausible.

What carries the argument

The central object is the turbulent Maxwell stress $M_{ij}$ of a tangled magnetic field, modeled as a viscoelastic fluid stress with a finite relaxation time $s$. The load-bearing term is the streamwise elastic stress, proportional to the second Rivlin-Eriksen tensor, which in this axisymmetric flow has only the azimuthal normal component; in a simplified near-equipartition treatment it is written $\nu s\gamma^2\hat\varphi\hat\varphi$, where $\gamma$ is the local shear rate and $\nu$ the turbulent viscosity. This term produces the first normal stress difference $N_1$ known to reverse secondary flows in laboratory viscometric flows. The mechanism acts through the identity $\nabla\times(\nabla\cdot(a\hat\varphi\hat\varphi))=-\partial_z(a/R)$ in cylindrical coordinates: for the Sun's observed shear, which decreases with $z$, this creates positive meridional vorticity and hence a clockwise circulation that counteracts the centrifugal driving. The polymer analogy carries the argument: a tangled field line under shear stretches and aligns like a polymer, giving an anisotropic tension, and a mean tension can exist without a mean field because the stress is quadratic in $B$.

Extended reading notes

Core claim

The central claim is a hypothesis: the presence of a magnetic field may make the calculated angular velocity profile more nearly coincide with the observed profile, and be an important ingredient in the solution of the Taylor number puzzle. The paper argues that the relevant physics lies not in the mean field but in the anisotropic, small-scale turbulent magnetic field, whose mean stress is viscoelastic in form and can be nonzero even when the mean field vanishes. In steady shear it produces a positive first normal stress difference, represented by an elastic streamwise stress $\nu s\gamma^2$ along $\hat\varphi$, a term proportional to the second Rivlin-Eriksen tensor. In cylindrical coordinates the curl of the divergence of such a stress is $-\partial_z(a/R)$; with the observed helioseismic shear profile this is a source of positive vorticity in the northern mid-latitudes, driving a clockwise circulation that opposes the counterclockwise circulation produced by the centrifugal term $R\partial_z\Omega^2$. The paper estimates the required streamwise-to-viscous stress ratio as $s\gamma\simeq10^2$ and the corresponding azimuthal Alfvén speed as a few times $10^4$ cm/s, i.e. fields of roughly $10^{1.5}$ kG at $\rho\sim0.05$ g/cm$^3$, and notes that MRI shearing-box simulations give ratios within one and a half orders of magnitude of this value.

Load-bearing premise

The whole mechanism depends on tangled magnetic fields in the Sun behaving like a polymer-like elastic fluid under shear, so that they pull back along their own length and create a tension that ordinary turbulence models leave out; if that behavior is absent, the proposed counter-flow does not exist.

Editorial extensions

If this is right

Mean-field models of the solar convection zone that include the viscoelastic turbulent Maxwell stress should reproduce the observed differential rotation profile at the realistic Taylor number $Ta\sim10^6$, without lowering $Ta$ by hand.
The required toroidal field strengths, of order $10^{1.5}$ kG at mid-convection-zone density, are plausible enough that the mechanism survives existing constraints on internal solar magnetic fields.
The predicted flow is a slowing, and possibly partial reversal, of the inertially driven meridional circulation in the mid-latitudes, giving a specific signature for helioseismic flow measurements.
Because the stress is quadratic in the fluctuating field, it can affect the angular momentum balance even where the net large-scale magnetic field vanishes, widening the range of solar models in which the mechanism operates.

Reading between the lines

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

If this mechanism operates, the strength of the balancing stress should track the magnetic energy of the convection zone, so the meridional circulation speed and possibly its sense should vary with the solar cycle; long helioseismic records could test this without further theory.
A parameter-free test would be to measure the first normal stress difference in shearing-box magnetohydrodynamic turbulence at solar parameters, since the closure requires $N_1/(\nu\gamma)\simeq s\gamma\sim10^2$, a number simulations can compute without assuming the viscoelastic analogy.
The same stress term could bear on the analogous rotation-profile problem in other stars with convective envelopes, where asteroseismology now constrains the surface differential rotation; the author does not pursue that extension.

Original abstract

Previous work on the angular momentum balance and meridional circulation of the solar convective zone (SCZ) generally consists either of semi-analytic approaches in which a simple turbulence model is adopted, or full direct numerical simulation (DNS) of the hydrodynamics or magnetohydrodynamics (MHD). In both instances the inclusion of magnetic fields has been troublesome. Also, both approaches have had difficulty reproducing the known angular velocity profile of the SCZ; this is the Taylor Number Puzzle. I discuss preliminary work in which I incorporate magnetic fields into a viscoelastic turbulence model for the SCZ and apply this to the problem of meridional circulation and angular momentum balance. I suggest that such an approach may help solve the Taylor Number Puzzle of the SCZ and bring theoretical predictions for the large-scale motion of the SCZ in line with observations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,342 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.75 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.58 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Previous work on the angular momentum balance and meridional circulation of the solar convective zone (SCZ) generally consists either of semi-analytic approaches in which a simple turbulence model is adopted, or full direct numerical simulation (DNS) of the hydrodynamics or magnetohydrodynamics (MHD). In both instances the inclusion of magnetic fields has been troublesome. Also, both approaches have had difficulty reproducing the known angular velocity profile of the SCZ; this is the Taylor Number Puzzle. I discuss preliminary work in which I incorporate magnetic fields into a viscoelastic turbulence model for the SCZ and apply this to the problem of meridional circulation and angular momentum balance. I suggest that such an approach may help solve the Taylor Number Puzzle of the SCZ and bring theoretical predictions for the large-scale motion of the SCZ in line with observations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Previous work on the angular momentum balance and meridional circulation of the solar convective zone (SCZ) generally consists either of semi-analytic approaches in which a simple turbulence model is adopted, or full direct numerical simulation (DNS) of the hydrodynamics or magnetohydrodynamics (MHD).	background 1.00	0.03	0.02	asserted 1.00
s1	In both instances the inclusion of magnetic fields has been troublesome.	background 0.59	0.05	0.02	asserted 1.00
s2	Also, both approaches have had difficulty reproducing the known angular velocity profile of the SCZ; this is the Taylor Number Puzzle.	background 0.60	0.05	0.04	asserted 1.00
s3	I discuss preliminary work in which I incorporate magnetic fields into a viscoelastic turbulence model for the SCZ and apply this to the problem of meridional circulation and angular momentum balance.	contribution 0.92	0.92	0.08	asserted 0.90
s4	I suggest that such an approach may help solve the Taylor Number Puzzle of the SCZ and bring theoretical predictions for the large-scale motion of the SCZ in line with observations.	interpretation 0.79	0.20	0.65	hedged possibility 1.00

What kind of paper the classifier reports this is: position or commentary 0.51 · evidence from argument 0.55.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (17)

[1]

Brandenburg A., Moss D., R \" u diger G., Tuominen I., 1990, Solar Physics, 128, 243

View on Pith 1990
[2]

Kitchatinov L. L., R \" u diger G., 1995a, A&A, 299, 446

View on Pith
[3]

Kitchatinov L. L., R \" u diger G., 1995b, Astron. Lett., 21, 191

View on Pith
[4]

Kitchatinov L. L., R \" u diger G., Khomenko G. A., 1994, A&A, 287, 320

View on Pith 1994
[5]

Longcope D. W., McLeish T. C. B., Fisher G. H., 2003, ApJ, 599, 661

View on Pith 2003
[6]

Meisch M. S., Brun A. S., Toomre J., 2006, ApJ, in press

View on Pith 2006
[7]

Ogilvie G. I., 2001, , 325, 231, astro-ph/0102245

View on Pith Pith review arXiv 2001
[8]

Ogilvie G. I., 2003, , 340, 969, astro-ph/0212442

View on Pith Pith review arXiv 2003
[9]

Rempel M., 2005, , 622, 1320

View on Pith 2005
[10]

R \" u diger G., 1989, Differential Rotation and Stellar Convection: Sun and Solar-type Stars. Gordon and Breach, New York

View on Pith 1989
[11]

Schou J., et al. 1998, ApJ, 505, 390

View on Pith 1998
[12]

Rekowski B. v., R \" u diger G., 1998, A&A, 335, 679

View on Pith 1998
[13]

Williams P. T., 2001, astro-ph/0111603

View on Pith Pith review arXiv 2001
[14]

Williams P. T., 2004, New Astr, 10, 133, astro-ph/0212556

View on Pith Pith review arXiv 2004
[15]

Williams P. T., 2005, MNRAS, 361, 345, astro-ph/0506184

View on Pith Pith review arXiv 2005
[16]

Yamaguchi H., Fujiyoshi F., Matsui H., 1997, J. Non-Newtonian Fluid Mech, 69, 29

View on Pith 1997
[17]

Yamaguchi H., Matsui H., 1997, J. Non-Newtonian Fluid Mech, 69, 47

View on Pith 1997

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Brandenburg A., Moss D., R \" u diger G., Tuominen I., 1990, Solar Physics, 128, 243

View on Pith 1990

[2] [2]

Kitchatinov L. L., R \" u diger G., 1995a, A&A, 299, 446

View on Pith

[3] [3]

Kitchatinov L. L., R \" u diger G., 1995b, Astron. Lett., 21, 191

View on Pith

[4] [4]

Kitchatinov L. L., R \" u diger G., Khomenko G. A., 1994, A&A, 287, 320

View on Pith 1994

[5] [5]

Longcope D. W., McLeish T. C. B., Fisher G. H., 2003, ApJ, 599, 661

View on Pith 2003

[6] [6]

Meisch M. S., Brun A. S., Toomre J., 2006, ApJ, in press

View on Pith 2006

[7] [7]

Ogilvie G. I., 2001, , 325, 231, astro-ph/0102245

View on Pith Pith review arXiv 2001

[8] [8]

Ogilvie G. I., 2003, , 340, 969, astro-ph/0212442

View on Pith Pith review arXiv 2003

[9] [9]

Rempel M., 2005, , 622, 1320

View on Pith 2005

[10] [10]

R \" u diger G., 1989, Differential Rotation and Stellar Convection: Sun and Solar-type Stars. Gordon and Breach, New York

View on Pith 1989

[11] [11]

Schou J., et al. 1998, ApJ, 505, 390

View on Pith 1998

[12] [12]

Rekowski B. v., R \" u diger G., 1998, A&A, 335, 679

View on Pith 1998

[13] [13]

Williams P. T., 2001, astro-ph/0111603

View on Pith Pith review arXiv 2001

[14] [14]

Williams P. T., 2004, New Astr, 10, 133, astro-ph/0212556

View on Pith Pith review arXiv 2004

[15] [15]

Williams P. T., 2005, MNRAS, 361, 345, astro-ph/0506184

View on Pith Pith review arXiv 2005

[16] [16]

Yamaguchi H., Fujiyoshi F., Matsui H., 1997, J. Non-Newtonian Fluid Mech, 69, 29

View on Pith 1997

[17] [17]

Yamaguchi H., Matsui H., 1997, J. Non-Newtonian Fluid Mech, 69, 47

View on Pith 1997