Modelling solar and stellar differential rotation - IOPscience

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Modelling solar and stellar differential rotationTo cite this article Manfred Kuumlker and Guumlnther Ruumldiger 2008 J Phys Conf Ser 118 012029

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This content was downloaded from IP address 451729944 on 01122021 at 2359

Modelling solar and stellar differential rotation

Manfred Kuker Gunther Rudiger

Astrophysikalisches Institut Potsdam An der Sternwarte 16 14482 Potsdam Germany

E-mail mkuekeraipde

Abstract We present a model of stellar differential rotation based on the mean-field theoryof fluid dynamics DR is driven by Reynolds stress and anisotropic heat transport caused bythe Coriolis force Our model reproduces the rotation pattern in the solar convection zone andallows predictions for other stars with outer convection zones We present results for a rangeof spectral types including the limiting case of very shallow convection zones and discuss thedependence of DR on the rotation rate and spectral type for main sequence stars

1 Introduction

Differential rotation (DR) is a powerful generator of magnetic fields and therefore a key ingredientin stellar dynamo models The surface DR of the sun has been known for a long time from thetracking of sunspots The rotation period at the solar equator is about 30 percent shorter thanthat at the poles Helioseismology has revealed that this pattern persists throughout the entireconvection zone while the radiative core rotates rigidly Between the core and the convectionzone there is a transition layer with strong radial shear [1] which is often called the tachoclineSurface DR is also found for other main-sequence stars Photometry of spotted stars shows avariation of the rotation period with the stellar activity cycle [2] Such stellar butterfly diagrams(in analogy to the solar butterfly diagram) give a lower estimate of the surface DR but can notdistinguish between solar-type and anti-solar rotation without additional information about thestarrsquos activity cycle [3] The Mt Wilson Ca HK project which monitors stellar Ca II activityfound butterfly diagrams similar to those from photometry [4] Spectroscopic measurement ofDR with the Fourier transform method can distinguish between solar-type and anti-solar rotationbut requires fast rotation and high luminosity Consequently it has been carried out for starsof spectral types A and F only [5] Doppler imaging derives the DR from the motions of surfacespots [6] Like the Fourier transform method it needs fast rotation Space-based photometryusing the MOST satellite has determined the DR of the stars ε Eri and κ1 Ceti [7 8]

So far no MS star has been found showing anti-solar DR ie a polar cap rotating with ashorter period than the stellar equator Stellar DR is usually characterized by the surface shear

δΩ = Ωeq minus Ωpole (1)

where Ωeq and Ωpole are the rotation rates at the equator and the poles respectively The surfaceshear is related to the lapping time tlap = 2πδΩ For stars a surface rotation law of the form

Ω = Ωeq(1 + k cos2 θ) (2)

Proceedings of the Second HELAS International Conference IOP PublishingJournal of Physics Conference Series 118 (2008) 012029 doi1010881742-65961181012029

ccopy 2008 IOP Publishing Ltd 1

is usually assumed where θ is the colatitude With that type of rotation law we have

δΩ =2π|k|

Prot (3)

where Prot is the rotation period at the equator While rotation laws of the form (2) are widelyused for fitting observation data it should be noted that in the rotation law derived from Dopplershifts for the solar surface [9]

Ω(θ) = (14050 minus 1492 cos2 θ minus 2606 cos4 θ)degday (4)

the cos4 θ term exceeds the cos2 θ term Rotation laws derived from the observation of sunspotshave k = minus02 [10 11] which is somewhat smaller than the equator-pole difference from theDoppler shifts and corresponds to δΩ = 005 radday and a lapping time of 135 days From therotation law (4) we find δΩ = 007 and tlap = 90 days

While observations so far show no systematic dependence of δΩ on the stellar rotation perioda temperature dependence of the form

δΩ prop T 892plusmn031 (5)

has been found [12] Theory thus not only has to explain the rotation pattern found in the solarconvection zone but also the variation of stellar DR along the main sequence

2 Model

DR can be explained with angular momentum transport by the convective gas motions Themean-field approach of magnetohydrodynamics treats the very complex gas motion in a stellarconvection zone by applying an average and solving the equation of motion for the mean gasmotion only In that equation the small-scale motions (ie the convection pattern and allmotions on scales smaller than that) only appear through a correlation tensor which acts asan additional stress The latter is called the Reynolds stress and depends on certain statisticalproperties of the gas motion only [13]

Our model combines DR meridional flow and convective heat transport The equation ofmotion for the mean gas flow in a stellar convection zone reads

ρ

[

partu

partt+ (u middot nabla)u

]

= nabla(π + R) minusnablaP + ρg (6)

where u is the mean velocity field P the gas pressure ρ the gas density and g the localacceleration due to gravity In first term on the RHS π and R are the molecular and Reynoldsstress tensors respectively

The convective heat transport is described by the equation

ρTpartδs

partt= minusnabla middot (F conv + F rad + ρT uδs) (7)

with δs defined asδs = s minus s0 δs ≪ s s0 (8)

In Eq 8 s is the mean entropy and the constant value s0 describes perfectly adiabaticstratification of the reference state [14] The radiative heat flux is given by

F radi = minus

16σT 3

3κρnablaiT (9)


2

with the Stefan-Boltzmann constant σ and the opacity κ The convective heat transport isdescribed by the flux vector

F convi = ρcp〈u

primeiT

prime〉 (10)

where uprime and T prime denote the fluctuations of velocity and temperature respectively The correlation〈uprime

iTprime〉 can be rewritten in terms of mean quantities

〈uprimeiT

prime〉 = χtΦijβj (11)

where χt is a scalar diffusion coefficient and Φij a dimensionless tensor [15] The diffusioncoefficient is determined by the stratification of the convection zone

χt = τcgα2H2p 〈βr〉12T (12)

where τc is the turbulent eddy correlation time α the classical mixing-length parameter Hp thepressure scale height 〈βr〉 the horizontally-averaged radial component of the super-adiabaticgradient

β = gcp minus nablaT (13)

and cp the specific heat capacity at constant pressure The tensor Φij depends on the Coriolisnumber

Ωlowast = 2πτcΩ (14)

which is closely related to the Rossby number

Ro =Prot

τc

=4π

Ωlowast (15)

For slow rotation Ωlowast ≪ 1 Φij is reduced to the Kronecker δ and χt is the same as in standardmixing length theory For fast rotation Ωlowast ≫ 1 both the magnitude of the diffusion coefficientand the structure of the tensor Φij The convective heat flux is then no longer aligned with thetemperature gradient Instead it is tilted towards the rotation axis so that there is a horizontalheat flux from the equator to the poles

The stress tensor Rij = minusρQij is determined by the gas density and the one-point correlationtensor of the velocity fluctuations Qij = 〈uprime

iuprimej〉 It can be expressed in terms of the mean gas

motion in which case it takes the form

Qij = minusNijkl

partΩk

partxl

+ ΛijkΩk (16)

The first term is zero for rigid rotation In the equation of motion it acts purely diffusive andthus represents a viscosity The second term the Λ-effect exists even in case of rigid rotationand therefore can cause DR

Like the convective heat transport we write the viscosity tensor as a product of a scalarfunction and a dimensionless tensor

Nijkl = νtΨijkl (17)

where νt = τcgα2H2p 〈βr〉15T and Ψijkl(Ω

lowast) dimensionless functions of the Coriolis number [15]In spherical polar coordinates the Λ-effect only appears in two components of the correlationtensor

QΛrφ = νtV sin θΩ V = V (0) + V (1) sin2 θ (18)

QΛθφ = νtH cos θΩ H = V (1) sin2 θ (19)


3

005 050 095 Fractional radius

0

25bull106

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Co

nve

ctive

tu

rno

ve

r tim

e [

s]


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Co

nve

ctive

tu

rno

ve

r tim

e [

s]

095 097 099 Fractional Radius

0

1bull104

2bull104

3bull104

Co

nve

ctive

Tu

rno

ve

r Ti

me

[s]

Figure 1 The convective turnover time as a function of depth for an M dwarf the sun anda 14 M⊙ main sequence star (from left to right) The rotation periods are 5 d 27 d and 1drespectively

with the dimensionless scalar functions V (0)(Ωlowast) and V (1)(Ωlowast) [16] For isotropic homogeneousfluctuations the Λ-effect vanishes The Reynolds stress is then reduced to usual turbulenceviscosity A stellar convection zone however is stratified and and rotates The gas motions aretherefore neither isotropic nor homogeneous and the Λ-effect appears in the stress tensor Forslow (but non-zero) rotation angular momentum is mainly transported in the radial directionwhile for fast rotation the horizontal effect dominates especially in thin convection zones

Figure 1 shows the convective turnover time vs the fractional stellar radius for three typesof main sequence star The left diagram shows a low-mass star with 035 solar masses rotatingwith a period of 5 d The diagram at the center shows the sun and the right diagram a mainsequence star with 14 solar masses rotating with a period of 1 d The low mass star has thethe longest convective time much longer than its rotation period The Coriolis number is thuslarge For the sun the Coriolis number is of the same order as the rotation period In the outerparts of the convection zone the rotation period is shorter than τc while in the bulk and at thebottom it is longer Hence the supergranulation layer is in a state of slow rotation the rest ofthe convection zone in a state of fast rotation The convective turnover time in the F-dwarf isshorter than in the bulk of the solar convection zone by almost two orders of magnitude Thisis a consequence of the small depth of this starrsquos convection zone which resembles the solarsupergranulation layer Because of the short convective time scale the star is in a state of slowrotation (Rogt1) even for a rotation period of 1 d

3 Results

The first application of our model is the solar rotation Figure 2 shows the resulting rotation andmeridional flow patterns As the model contains the outer convection zone only the tachoclineis missing Nevertheless the rotation shows the main characteristics of the solar DR namelya fast equator a slow pole and little variation with radius in the bulk of the convection zoneBoth the surface flow and the horizontal shear are remarkably close to the observed values Themeridional flow shows one flow cell per hemisphere with the surface flow directed towards thepoles and the bottom flow towards the equator The amplitude is about 16 ms at the top and 8ms at the bottom Note that the return flow is confined to a shallow layer at the bottom of theconvection zone The difference between equatorial and polar rotation rates is about 25 percentA least-squares fit to a rotation law of the type (2) gives k = minus021 The corresponding valuesof δΩ are 005 radd and 006 rads respectively the lapping times are 119 and 100 days

Next we apply our model to a star on the lower main sequence Figure 3 shows the meridional


4

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Figure 2 Meridional flow and DR for the sun with a rotation period of 27d Left stream linesof meridional flow Blue indicates counter-clockwise flow red clockwise Center color contourplot of the normalized rotation rate Right the meridional flow speed at the top (blue) andbottom (red) of the convection zone

095

098

101

104

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90 45 0 -45 -90Latitude

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-2

0

2

4

6

ms

Figure 3 Meridional flow and DR for a fully-convective low-mass star Left stream lines ofmeridional flow Blue indicates counter-clockwise flow red clockwise Center color contour plotof the (normalized) rotation rate Right meridional flow speed at the top (blue) and bottom(red) of the convection zone


5

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091

096

101

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-200

-100

0

100

200

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Figure 4 Meridional flow and rotation of an F dwarf with 14 M⊙ Left stream lines ofmeridional flow Center color contour plot of the normalized rotation rate Right meridionalflow speed at the top (blue) and bottom (red) of the convection zone

flow and rotation of an M dwarf rotating with a period of 5 d As in the solar convection zonethere is one flow cell per hemisphere with the surface flow directed towards the poles The flowamplitude at the surface is 6 ms A small core with a radius of 5 percent of the stellar radius ispresent for numerical reasons The color contour plot in Fig 3 shows little variation expect closeto the rotation axis Variation inside the cylinder surrounding the core is an artifact caused bythe boundary condition For the equator-pole difference of the rotation period we find a valueof 28 percent of the equatorial rotation rate or a total of 0035 radd A fit to a cos2 θ law oftype (2) yields k = minus0028 and 0035 radd The lapping time is 180 d Despite the fact that therotation profile appears much more rigid in the color contour plot the total shear δΩ is morethan half the solar value The rotation pattern is much closer to the Taylor-Proudman statethough with Ω increasing with radius in the bulk of the convection zone

At the other end of the lower main sequence we have the F-dwarf with 14 solar masses Thedepth of the convection zone in this star is only five percent of the stellar radius Figure 4 showsthe meridional flow and rotation patterns for this star at a rotation period of 1 d In the left andcenter diagrams the depth of the convection zone has been artificially increased by a factor offive for clarity The meridional flow is very fast in comparison with the sun There is one flowcell per hemisphere with the surface flow directed towards the poles The maximum flow speedis 140 ms at the surface and 100 ms at the bottom The rotation pattern is similar to thesolar rotation with an equator-pole difference of 17 percent of the equatorial rotation rate Forthe total shear we find δΩ = 104 radd corresponding to a lapping time of only 6 d From thecos2 θ fit we get k = minus02 δΩ = 125 radd and a lapping time of 5 d

Figure 5 summarizes observations and models for a number of stars In the left diagram thelines denote the total surface shear δΩ as a function of the rotation period for four types of mainsequence star from F8 to M2 The triangles show observed surface DR from Doppler imagingfor four rapidly rotating young stars The diamonds show the sun and the stars ǫ Eri and κ1

Ceti DR of the latter two was derived from from their light curves as recorded by the MOSTsatellite The model shows a rather modest variation of the surface shear both with the rotationperiod and with spectral type Hotter stars have more DR than cool ones and fast rotators


6

Figure 5 Left differential rotation vs rotation period for MS stars The solid lines showthe surface shear for model stars of spectral types F8 G2 K5 and M2 The symbols indicateobservations The triangles denote results from Doppler imaging for the stars Speedy Mic ABDor and PZ Tel The diamonds in the left half of the diagram indicate surface DR derivedfrom photometry for ǫ Eri and κ1 Ceti The diamond in the right part marks the sun Rightdifferential rotation vs temperature The line indicates the T 892 law from [12] The diamondsdenote the results of model calculations In some cases the rotation period has been varied forthe same star leading to vertical scatter

have more shear than slow ones but both trends are rather weak So far we have not been ableto compute models for rotation periods as short as 05 days Extrapolating from the availablemodels to the rotation periods where Doppler imaging is possible we find very good agreementbetween theory and observations

The right diagram of Fig 5 shows the surface shear as a function of the effective temperatureThe line denotes the T 892 law of [12] the diagrams indicate results from our model for varioustypes of star and rotation periods In the right half of the diagram the diamonds scatter aroundthe curve in the left they lie above it This indicates a weaker temperature dependence thanfound by [12] In the right half on the other hand we find some quite large values for δΩespecially for the F-dwarf discussed above

4 Discussion

Based on the mean-field theory of hydrodynamics the second order correlation approximationand the mixing length theory of stellar convection we have constructed models for the rotationand meridional flow of main sequence stars with outer convection zones This model successfullyreproduces the rotation pattern of the solar convection zone and the meridional flow It alsoallows predictions for other stars For a given type of star the dependence of the surface shearon the rotation period is weak Variation with spectral type is moderate at the lower end ofthe MS but steep for effective temperatures above asymp 6500 K For masses above 14M⊙ theconvection zone is very shallow and the convective time scale is short These objects musttherefore rotate fast for convection to be significantly affected by rotation On the other handthe Fourier transform method can only detect differential rotation if k gt 01 Together with theshort rotation periods required this results in large values of δΩ for stars with observed surfaceDR Stars with weaker DR or slow rotators would not be detected as differential rotators In ourmodel the large values of δΩ for stars with masses above 14 solar masses are the result of the


7

short convective time scales in the shallow convection zones of these stars They require veryfast rotation in order to reach Ωlowast asymp 1 In a star with 14 solar masses rotating with an averageperiod of 27 d the Coriolis number would be as small as 016 at the bottom of the CZ and theimpact of rotation on the convection pattern weak The resulting surface shear would be smalland not be detected by the Fourier transform method

References[1] Thompson MJ Toomre J Anderson E et al 1996 Science 272 1300[2] Henry GW Eaton JA Hamer J Hall DS 1995 Astrophys J Suppl 97 513[3] Messina S Guinan EF 2003 Astron amp Astrophys 409 1017[4] Donahue RA Saar SH Baliunas SL 1996 Astrophys J 466 384[5] Reiners A 2006 Astron amp Astrophys 446 267[6] Collier Cameron A 2002 Astron Nachr 323 336[7] Croll B Walker G Kuschnig R et al 2006 Astrophys J 648 607[8] Walker G Croll B Kuschnig R et al 2007 Astrophys J 659 1611[9] Snodgrass HP 1984 Solar Phys 94 13S[10] Howard R Gilman PI Gilman PA 1984 Astrophys J 283 373[11] Balthasar H Lustig G Stark D Wohl H 1986 Astron amp Astrophys 160 277[12] Barnes JR Collier Cameron A Donati J-F et al 2005 Month Not Roy Astron Soc 357 L1[13] Rudiger G Hollerbach R 2004 The Magnetic Universe Geophysical and Astrophysical Dynamo Theory

Wiley-VCH Weinheim[14] Bonanno A Kuker M Paterno L 2006 Astron amp Astrophys 462 1031[15] Kitchatinov LL Pipin VV Rudiger G 1994 Astron Nachr 315 157[16] Kitchatinov LL Rudiger G 1994 Astron amp Astrophys 276 96


8

Modelling solar and stellar differential rotation

Manfred Kuker Gunther Rudiger

Astrophysikalisches Institut Potsdam An der Sternwarte 16 14482 Potsdam Germany

E-mail mkuekeraipde

Abstract We present a model of stellar differential rotation based on the mean-field theoryof fluid dynamics DR is driven by Reynolds stress and anisotropic heat transport caused bythe Coriolis force Our model reproduces the rotation pattern in the solar convection zone andallows predictions for other stars with outer convection zones We present results for a rangeof spectral types including the limiting case of very shallow convection zones and discuss thedependence of DR on the rotation rate and spectral type for main sequence stars

1 Introduction

Differential rotation (DR) is a powerful generator of magnetic fields and therefore a key ingredientin stellar dynamo models The surface DR of the sun has been known for a long time from thetracking of sunspots The rotation period at the solar equator is about 30 percent shorter thanthat at the poles Helioseismology has revealed that this pattern persists throughout the entireconvection zone while the radiative core rotates rigidly Between the core and the convectionzone there is a transition layer with strong radial shear [1] which is often called the tachoclineSurface DR is also found for other main-sequence stars Photometry of spotted stars shows avariation of the rotation period with the stellar activity cycle [2] Such stellar butterfly diagrams(in analogy to the solar butterfly diagram) give a lower estimate of the surface DR but can notdistinguish between solar-type and anti-solar rotation without additional information about thestarrsquos activity cycle [3] The Mt Wilson Ca HK project which monitors stellar Ca II activityfound butterfly diagrams similar to those from photometry [4] Spectroscopic measurement ofDR with the Fourier transform method can distinguish between solar-type and anti-solar rotationbut requires fast rotation and high luminosity Consequently it has been carried out for starsof spectral types A and F only [5] Doppler imaging derives the DR from the motions of surfacespots [6] Like the Fourier transform method it needs fast rotation Space-based photometryusing the MOST satellite has determined the DR of the stars ε Eri and κ1 Ceti [7 8]

So far no MS star has been found showing anti-solar DR ie a polar cap rotating with ashorter period than the stellar equator Stellar DR is usually characterized by the surface shear

δΩ = Ωeq minus Ωpole (1)

where Ωeq and Ωpole are the rotation rates at the equator and the poles respectively The surfaceshear is related to the lapping time tlap = 2πδΩ For stars a surface rotation law of the form

Ω = Ωeq(1 + k cos2 θ) (2)


ccopy 2008 IOP Publishing Ltd 1


δΩ =2π|k|

Prot (3)







2 Model



ρ

[

partu


]




ρTpartδs




F radi = minus

16σT 3

3κρnablaiT (9)


2


F convi = ρcp〈u

primeiT

prime〉 (10)



〈uprimeiT









Ro =Prot

τc

=4π

Ωlowast (15)





Qij = minusNijkl

partΩk

partxl

+ ΛijkΩk (16)









3


0

25bull106

50bull106

75bull106

Co

nve

ctive

tu

rno

ve

r tim

e [

s]


0

50bull105

10bull106

15bull106

Co

nve

ctive

tu

rno

ve

r tim

e [

s]


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2bull104

3bull104

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nve

ctive

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rno

ve

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[s]




3 Results




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4 Discussion



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8


δΩ =2π|k|

Prot (3)







2 Model



ρ

[

partu


]




ρTpartδs




F radi = minus

16σT 3

3κρnablaiT (9)


2


F convi = ρcp〈u

primeiT

prime〉 (10)



〈uprimeiT









Ro =Prot

τc

=4π

Ωlowast (15)





Qij = minusNijkl

partΩk

partxl

+ ΛijkΩk (16)









3


0

25bull106

50bull106

75bull106

Co

nve

ctive

tu

rno

ve

r tim

e [

s]


0

50bull105

10bull106

15bull106

Co

nve

ctive

tu

rno

ve

r tim

e [

s]


0

1bull104

2bull104

3bull104

Co

nve

ctive

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rno

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r Ti

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[s]




3 Results




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4 Discussion



7





8


F convi = ρcp〈u

primeiT

prime〉 (10)



〈uprimeiT









Ro =Prot

τc

=4π

Ωlowast (15)





Qij = minusNijkl

partΩk

partxl

+ ΛijkΩk (16)









3


0

25bull106

50bull106

75bull106

Co

nve

ctive

tu

rno

ve

r tim

e [

s]


0

50bull105

10bull106

15bull106

Co

nve

ctive

tu

rno

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r tim

e [

s]


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2bull104

3bull104

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nve

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rno

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[s]




3 Results




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s]


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15bull106

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nve

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tu

rno

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r tim

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s]


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3 Results




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Modelling solar and stellar differential rotation - IOPscience