Magnetic Processes in Astrophysics E-Book

Günther Rüdiger

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2013

Seitenzahl: 616

Beschreibung

In this work the authors draw upon their expertise in geophysical and astrophysical MHD to explore the motion of electrically conducting fluids, the so-called dynamo effect, and describe the similarities and differences between different magnetized objects. They also explain why magnetic fields are crucial to the formation of the stars, and discuss promising experiments currently being designed to investigate some of the relevant physics in the laboratory. This interdisciplinary approach will appeal to a wide audience in physics, astrophysics and geophysics. This second edition covers such additional topics as small-scale dynamos, while also presenting the latest results and experiments.

Leseprobe

Contents

Preface

1 Differential Rotation of Stars

1.1 Solar Observations

1.2 Stellar Observations

1.3 The Reynolds Stress

1.4 The Meridional Flow

1.5 The Sun

1.6 Individual Stars

1.7 Dwarfs & Giants

1.8 Differential Rotation along the Main Sequence

2 Radiation Zones: Magnetic Stability and Rotation

2.1 The Watson Problem

2.2 The Magnetic Tachocline

2.3 Stability of Toroidal Fields

2.4 Stability of Thin Toroidal Field Belts

2.5 Helicity and Dynamo Action

2.6 Ap Star Magnetism

2.7 The Shear–Hall Instability (SHI)

3 Quasi-linear Theory of Driven Turbulence

3.1 The Turbulence Pressure

3.2 The η-Tensor

3.3 Kinetic Helicity and DIV-CURL Correlation

3.4 Cross-Helicity

3.5 Shear Flow Electrodynamics

3.6 The Alpha Effect

3.7 The Current Helicity

4 The Galactic Dynamo

4.1 Magnetic Fields of Galaxies

4.2 Interstellar Turbulence

4.3 Dynamo Models

4.4 Magnetic Instabilities

5 The Magnetorotational Instability (MRI)

5.1 Taylor–Couette Flows

5.2 The Stratorotational Instability (SRI)

5.3 The Standard Magnetorotational Instability (SMRI)

5.4 Diffusive Kepler Disks

5.5 MRI with Hall Effect

5.6 The Azimuthal MRI (AMRI)

5.7 Helical Magnetorotational Instability (HMRI)

5.8 Laboratory Experiment PROMISE

6 The Tayler Instability (TI)

6.1 Stationary Fluids

6.2 Experiment GATE

6.3 Rotating Fluids

6.4 The Tayler Generator

6.5 Helical Background Fields and Alpha Effect

6.6 TI with Hall Effect

7 Magnetic Spherical Couette Flow

7.1 Stewartson Layers

7.2 Shercliff Layers

7.3 Finite Re in an Axial Field

7.4 The Grenoble DTS Experiment

7.5 Other Waves and Instabilities

7.6 Linear Combinations of Axial and Dipolar Fields

7.7 Dynamo Action

References

Index

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The Authors

Prof. Günther RüdigerLeibniz-Institut für Astrophysik Potsdam (AIP)An der Sternwarte 1614482 PotsdamGermany

Dr. Leonid L. KitchatinovInstitute for Solar-TerrestrialPhysiksLermontov st., 126 a664033 IrkutskRussian Federation

Prof. Rainer HollerbachInstitut für GeophysikETH Zürich8092 ZürichSwitzerland

Cover PictureThe PROMISE facility for experimental studies of the helical and the azimuthal magnetorotational instability, constructed and operated at Helmholtz-Zentrum Dresden-Rossendorf.

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Preface

In 2004 two of us (Rüdiger and Hollerbach) published a previous book entitled The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory, describing the origin of magnetic fields in objects ranging in size from planets to galaxies. Ever since then, we have considered the possibility of writing a second edition, updating developments of the past decade. However, ultimately there were so many recent developments in areas not covered at all before that it ended up as a completely new book, with only minimal overlap (in parts of Chapters 1 and 4) with corresponding formulations in Rüdiger and Hollerbach (2004). In particular, the subjects of these two chapters (differential rotation theory, the galactic dynamo) have developed so rapidly since then that a new discussion was clearly necessary.

On the one hand, the successful asteroseismic space missions MOST, COROT and KEPLER ushered in a new era of knowledge of the internal stellar rotation laws. Stars exhibit much greater variety of turbulent convection zones and angular momentum than found in the Sun, so that it is now possible to develop the theory of the rotation of stars by means of the new data. It is also clear that only facts about stellar differential rotation allow us to understand the magnetic activity of the main-sequence stars.

One significant change though from Rüdiger and Hollerbach (2004) is the switch in emphasis from being primarily on dynamo theory more toward magnetic instabilities such as the magnetorotational instability or the Tayler instability. That is, instead of seeking to explain the origin of magnetic fields, we now take the existence of large-scale fields as given, and study the ways in which instabilities can destroy the large-scale structures again, and give rise to small-scale turbulence instead.

Another aspect that has changed significantly since 2004 is the increasing importance of liquid metal (and plasma) laboratory experiments, not only in attempts to create laboratory dynamos, but also involving externally imposed magnetic fields. In addition to the chapters on the magnetorotational and Tayler instabilities, the chapter on magnetic spherical Couette flow describes a number of new experiments. In all of these areas, the interplay between basic theory, detailed numerical simulations, and experiments has been particularly fruitful, with generally good agreement also between theory and experiment (perhaps “disappointingly” good, if one views unexpected experimental results as the ones most likely to further lead to fundamentally new insights). At any rate, we hope that this book will be of interest not just to astrophysicists but to fluid dynamicists more generally, or anyone else wanting to understand liquid metal experiments and the insights they can yield.

Numerous colleagues have contributed to this book, either directly or by general discussions over many years. GR particularly thanks Rainer Arlt, Detlef Elstner, Marcus Gellert, Andrea Hans, Manfred Küker and Manfred Schultz of the Leibniz-Institut für Astrophysik Potsdam for their substantial support with countless technical details. LLK and GR acknowledge the continuous encouragement of the Deutsche Forschungsgemeinschaft and the Alexander von Humboldt Foundation stimulating a number of the developments presented here. RH thanks Prof Andy Jackson of the Institute of Geophysics at ETH Zürich for the invitation to visit ETH (with funding by the European Research Council). The time away from regular duties was invaluable in finishing this project in time. Finally, among the vast MHD community we particularly thank Gunter Gerbeth, Thomas Gundrum, Martin Seilmayer, Frank Stefani and the entire group at the Helmholtz-Zentrum Dresden-Rossendorf for many intensive discussions and collaborations over the past decade, as presented in several of the chapters here, and which we hope may stimulate further developments in MHD laboratory astrophysics.

PotsdamIrkutskZürich2013

Günther RüdigerLeonid L. KitchatinovRainer Hollerbach

1

Differential Rotation of Stars

Magnetic activity of solar-type stars is closely related to stellar rotation. The differential rotation participates in stellar dynamos by producing toroidal magnetic fields by rotational shear. Differential rotation and meridional flow can be understood in the context of mean-field hydrodynamics in stellar convection zones. Stratification in convection zones is so strong that the Schwarzschild criterion (dS/dr < 0, where S is the specific entropy) is fulfilled and the entire zone becomes turbulent. Due to the radial stratification the turbulence fields are themselves stratified with the radial preferred direction. Interaction of such a turbulence with an overall rotation leads to the formation of large-scale structure. Lebedinskii (1941), Wasiutynski (1946), Biermann (1951) and Kippenhahn (1963) were the first to find that differential rotation and meridional flow might be direct consequences of the rotating anisotropic turbulence. Details of the long history of this concept were presented by Rüdiger (1989, Chapter 2).

Whether a star is of solar-type is controlled by its structure. Stars of this type possess external (turbulent) convection zones. The solar convection zone only includes < 2% of the total mass but it extends about 30% in radius. The outer convection zones in cooler stars become deeper as stellar mass decreases until for M stars the convection zone reaches down to the center. On the other hand, for A stars the outer convection zone becomes very thin, but an inner zone becomes convectively unstable. For B stars this inner convection zone reaches considerable dimensions.

The level of stellar activity depends strongly on spectral type. There is, however, the striking fact that the linear depth of the outer convection zone, at 200 000 km, does not vary too much among the solar-type stars. We shall see later how important the total thickness of a convection zone is for the formation of differential surface rotation.

It is certainly unrealistic to expect a solution of the complicated problem of stellar dynamos if the internal stellar rotation laws cannot be predicted or observed (by asteroseismology). Differential rotation is explained here as turbulence-induced with only a small magnetic contribution. Mean-field hydrodynamics provides a theoretical basis for differential rotation modeling, so that the models can be constructed with very little arbitrariness. Nevertheless, differential rotation of the Sun can be reproduced by computations very closely and the dependence of differential rotation on stellar parameters can be predicted.

1.1 Solar Observations

1.1.1 The Rotation Law

The rotation of the solar photosphere was measured using the Doppler shifts of photospheric spectral lines or tracking rotation of sunspots and various other tracers. Doppler measurements of Howard et al. (1983) and the classical work of Newton and Nunn (1951) on sunspot rotation are the well-known examples. Within a small percentage, all measurements yield similar results. Obtained by tracing bright coronal structures in SOHO images Wöhl et al. (2010) give

(1.1)

(1.2)

(here ≈ 0.23) because only ∇Ω is relevant for the inducting action of differential rotation but not its normalized value k. With (1.2) we follow the notation of the seminal paper by Hall (1991) who derived from photometric stellar observations a relation k ∝ Ω–0.85 (corresponding to the very flat relation δΩ ∝ Ω0.15 for rotating stars, see also Barnes et al. (2005)) which is rather close to the essentials presented in the theoretical part of this chapter.

Figure 1.1 Isolines of the angular velocity of the Sun after Korzennik and Eff-Darwich (2011). The rotation of the polar and the near-center regions is difficult to measure. With permission of the authors.

The main empirical features of the solar differential rotation can be summarized as follows (see Figure 1.1):

an equatorial acceleration of about 23% at the surface,

a near-surface shear layer with negative

Ω

-gradient in radius

1

)

,

a sharp transition to rigid rotation in a thin tachocline,

a rigid rotation of the deeper radiative core.

The ‘observed’ phenomenon of the sharp transition layer between the outer domain of differential rotation and the inner domain of rigid-body rotation is hard to understand without the assumption of internal empirically unknown magnetic fields. We shall show in Section 2.2 that indeed fossil fields with amplitudes of only 1 mG are enough to explain not only the existence of the tachocline but also its small radial extension.

The present state of differential rotation may, however, differ from other epochs when magnetic activity of the Sun was different. Ribes and Nesme-Ribes (1993) used statistics of sunspot observations over the Maunder minimum at the Observatoire de Paris to find a rotation rate slower by about 2% at the equator and by about 6% at midlatitudes than at the present time. The differential rotation was thus stronger than today. The more magnetic the Sun, the faster and more rigidly its surface rotates. Balthasar, Vázquez, and Wöhl (1986), however, could not find similar results for a regular minimum. Also Arlt and Fröhlich (2012), who worked with data obtained from the drawings of Staudacher from the period from 1749 till 1799 did not find a significant difference to the present-day value of δΩ 0.050 rad/day derived by Balthasar, Vázquez, and Wöhl (1986) from sunspot rotation. The reported average value of 0.048 indicates a slightly smaller value but this difference is not yet significant.

Figure 1.2 The butterfly diagram shortly after the Maunder minimum, as derived from the drawings of Staudacher between 1749 and 1799. Courtesy of R. Arlt.

The results are nevertheless highly interesting as they demonstrate the reliability of the data which also led to the construction of a butterfly diagram for the four cycles covered by the observations. The main question here is whether the dipolar parity which now dominates the solar activity already existed shortly after the Maunder minimum. This is certainly the case for the last two cycles shown in Figure 1.2 but it seems to be questionable for the older two cycles. For these cycles, which are closer to the Maunder minimum at least an overpopulation of near-equator sunspots is indicated by the data (Arlt, 2009).

1.1.2 Torsional Oscillations

As magnetic activity of the Sun varies with time, differential rotation may also be expected to be time-dependent. Variations of solar rotation law are indeed observed. Schrijver and Zwaan (2000), Stix (2002) and Thompson et al. (2003) presented detailed historical and data-based overviews of all phenomena concerning the temporal variations of the solar rotation law. As the magnetic force is quadratic in the magnetic field, the resulting flow is expected to vary with twice the frequency of the 22-year magnetic cycle. The 11-year torsional oscillations were first observed by Howard and LaBonte (1980).

Figure 1.3 shows the oscillation pattern. At a fixed latitude there is an oscillation of fast and slow rotation with an 11-year period. The whole pattern migrates at about 2 m/s toward the equator. The migration follows the equatorial drift of magnetic activity. Latitudinal shear of differential rotation is increased in the activity belt with faster than average rotation on the equatorial side of the belt and slower than average rotation on the polar side. Howe, Komm, and Hill (2002) showed by helioseismological inversions that the migrating torsional oscillation exists not only at the surface, but that it extends at least 60 000 km down into the convection zone.

Figure 1.3 Torsional oscillations derived from Doppler shift measurements. The flow pattern follows the equatorial drift of magnetic activity. The flow at a given latitude oscillates with a period of about 11 years. Courtesy of Howe et al. (2011).

Figure 1.4 Helioseismology detects two branches of torsional oscillations migrating to the equator and to the poles from midlatitude and extending deep into the convection zone. From Vorontsov et al. (2002). Reprinted with permission from AAS.

An important question of dynamo theory is whether a poleward migrating branch is present at high latitudes. Schou (2001) and Vorontsov et al. (2002) reported the detection of such a branch of torsional oscillation for the rising phase of solar cycle 23 from helioseismological data between 1996 and 2002. Figure 1.4 shows the polar branch together with the low-latitude equatorial branch penetrating deep into the convection zone.

Close correlation with solar activity is indicative of a magnetic origin of torsional oscillations. Details of the mechanism producing the oscillations remain, however, uncertain. The oscillations may be produced by the global Lorentz force (Yoshimura, 1981; Schüssler, 1981; Rüdiger et al., 1986) or the magnetic backreaction on small spatial scale of turbulence (Kitchatinov, 1990; Rüdiger and Kitchatinov, 1990). Meridional flow induced by entropy disturbances in the magnetic activity belt may also be relevant (Spruit, 2003; Cameron and Schüssler, 2012).

1.1.3 Meridional Flow

It was recognized since the work of Kippenhahn (1963) that differential rotation and meridional flow are closely related and it is not possible to correctly describe one if the other is not allowed for.

The relatively slow meridional circulation is difficult to measure. Ward (1965) noticed that the flow is problematic to define by the method of tracers: an inhomogeneity of tracer distribution over latitude together with latitudinal turbulent diffusion results in a false meridional flow. This is probably why early measurements using sunspots as tracers gave conflicting results. A more coherent picture is provided by using more uniformly distributed ‘small magnetic features’ as tracers. Komm, Howard, and Harvey (1993) found a meridional flow from the equator to pole with amplitude slightly above 10 m/s. The flow is shown in Figure 1.5. The flow pattern shows no hemispheric asymmetry and did not migrate in latitude during a solar cycle. The amplitude of the flow varies, however, over the activity cycle. Meridional velocity changes from below average during solar maximum to above average during solar minimum. The relative amplitude of the variation is about 25%. This picture is supported by recent measurements of Hathaway and Rightmire (2010) and by seismological sounding of the flow by Basu and Antia (2010).

Figure 1.5 Full line and symbols show the meridional flow measured by using small magnetic features as tracers. Overplotted are Doppler measurements of Ulrich (1993) (dashed-dotted) and Snodgrass (1984) (dashed). From Komm, Howard, and Harvey (1993).

Zhao and Kosovichev (2004) measured the meridional flow for seven Carrington rotations of years 1996–2002 covering the epoch from solar activity minimum to maximum. The measurements by time-distance helioseismology show the poleward flow decreasing with depth in the surface layer of 12000 km. In addition to the dominating poleward flow of order 20 m/s, cells of weaker flow converging to the activity belts were found in both hemispheres. These cells migrated towards the equator following the migration of activity belts as the solar cycle evolved. These migrating cells may be a counterpart of torsional oscillations in the meridional flow.

Figure 1.6 Helioseismology results (MDI) for the rotational velocity (a) and meridional flow (b) in their dependencies on latitude and time (from 1996 (blue) to 2002 (red)). The meridional flow is poleward with an amplitude of 10 m/s, which varies in time by 7 m/s. From Gizon and Rempel (2008).

The question for the equatorward return flow deep in the convection zone has also been considered by Hathaway et al. (2003). From the sunspot data since 1874 they found an anticorrelation between the drift rate of the center of the butterfly diagram and the cycle length. The faster the drift of the butterfly diagram the shorter the cycles. With such statistics an amplitude of 1.2 m/s for the return flow velocity at the bottom of the convection zone has been estimated.

1.2 Stellar Observations

1.2.1 Rotational Evolution

Rotation of a star is an important parameter of hydromagnetic dynamos controlling the rate of magnetic field generation. The dynamo activity in turn decreases the rotation rate. Solar-type stars are observed to exhibit a steady decline in rotation rate between the ages from about 108 to 4.5 × 109 years (Skumanich, 1972).

The spin-down is commonly explained as follows (Kraft, 1967): magnetic activity of stars with external convection zones produce hot stellar coronae. Similar to the Sun, the hot coronae emanate stellar winds. The material making up the wind does not lose contact with the parent star after leaving its photosphere, but corotates with the star due to magnetic coupling to its surface. The extent of the coupling can be (very crudely) estimated by the Alfvén radius, RA, where the wind velocity equals the Alfvén speed. As the angular momentum loss is proportional to R2A, magnetic activity enhances the rotational braking. Spindown of the main-sequence dwarfs closely obeys the Skumanich law,

(1.3)

relating rotation period Prot of a star to its age t.

This law does not, however, apply to all stellar ages. Solar mass stars are born with rotation periods of about one week. Subsequently, these stars spin-up very quickly during contraction to the main-sequence to attain a rotation period of about one day or even shorter as ZAMS stars (Hartmann and Noyes, 1987). Close to the end of their main-sequence lives, stars seem to deviate from relation (1.3) as well. Figure 1.8 shows that the upper-left corner on the plot of Prot vs. B-V color index is empty. This suggests that the dwarf stars are not decelerated beyond a maximum rotation period depending on the spectral type. The maximum period is larger for cooler stars.

The spin-down law (1.3) applies to the solar-type stars over a major part of their main-sequence life. Gray (1982) and Rengarajan (1984) forwarded the idea that the proportionality constant in the relation (1.3) is a single-valued function of stellar mass or other equivalent parameter. This idea eventually led to the development of gyrochronology establishing an empirical relation between age, rotation period and mass (Barnes, 2003, 2007, 2010; Collier Cameron et al., 2009; Meibom, Mathieu, and Stassun, 2009). The relation is illustrated by Figure 1.9 where the dependence of the ratio Prot/ on B-V color is shown. Different functional expressions for the empirical relation have been suggested, for example

Figure 1.8 Plot of Prot (in days) versus B-V color for main-sequence stars. The squares and triangles refer to young and old stars, respectively. From Rengarajan (1984). Reprinted by permission of AAS.

Figure 1.9 The ratio of Prot (days) to square root of t (Myr) in dependence on B-V color for main-sequence stars of the Mount Wilson sample. Small and large circles show young and old stars, respectively. From Barnes (2007). Reprinted by permission of AAS.

(1.4)

1.2.2 Differential Rotation

Stellar differential rotation is measured mainly by the same method, which originally was used for the Sun, that is, by tracing the rotation of thermal or magnetic spots. As stars are typically point sources (only in very rare, exceptional cases can stellar surfaces be resolved), the methods are very sophisticated and demanding of observational data. The tracer method can be realized using high precision (space-based) photometry or high resolution spectroscopy with the Doppler imaging techniques. The differential rotation was also measured using shapes of spectral lines (Reiners and Schmitt, 2003a,b) and variations of Ca II H&K emissio (e.g., Donahue, Saar, and Baliunas, 1996). Doppler imaging (Khokhlova, 1975; Vogt and Penrod, 1983) provides detailed mapping of stellar surfaces but can be used only for young rapidly rotating stars because the projected rotation velocity v sin i should typically not be smaller than 15km/s. Measurements of differential rotation by Doppler imaging were summarized by Barnes et al. (2005) to reveal strong dependence on spectral type: the hotter the star, the larger the pole–equator difference in rotation rate. Observed differential rotation is usually fit by the sin2b profile

Figure 1.10 Dependence of surface differential rotation on effective temperature for rapidly rotating solar analogs by Doppler imaging (Barnes et al., 2005). Copyright © 2005 RAS.

(1.5)

Figure 1.10 shows the pole–equator difference δΩ as a function of temperature for young solar-type stars. The largest differential rotation belongs to the hottest stars. Jeffers and Donati (2008) found the slightly premain-sequence G0 star HD 171488 (V889 Her) with δΩ 0.5 rad/day exceeding all stars of Figure 1.10 in magnitude of its differential rotation (see Section 1.6.2).

When common statistics for stars of different spectral types is used, however, an increase of differential rotation with rotation rate is usually found (cf. Donahue, Saar, and Baliunas (1996)). As can be seen from Figure 1.8 or the gyrochronology equation (1.4), slow rotators are mainly represented by K stars while G and F stars show much shorter rotation periods. The increase of differential rotation with rotation rate is found because rotation of cooler stars is more uniform (Figure 1.10).

Figure 1.11 Observed differential rotation of stars close to the Sun by mass but rotating with different rates. LQ Lup is a premain-sequence star (Donati et al., 2000). AB Dor (Donati and Collier Cameron, 1997), PZ Tel (Barnes et al., 2000) and LQ Hya (Donati et al., 2003) are ZAMS stars. Flores-Soriano and Strassmeier (2013) give δΩ 0.04 rad/day for LQ Hya. Differential rotation of these stars was measured by Doppler imaging. High-precision photometry of the Most mission was used for older stars ε Eri (Croll et al., 2006) and k1 Ceti (Walker et al., 2007).

For the very young T Tau stars the early investigations led to almost solid-body rotation. Rice and Strassmeier (1996) found only a ratio δΩ/Ω 0.001 for V410 Tau with its rotation period of 1.87 days. The equator–pole difference results in the positive but small value δΩ 0.0035 rad/day. In a recent analysis of Most data Siwak et al. (2011) confirmed the smallness of this value (δΩ 0.002 rad/day) but for two other weak-line TTS the values δΩ 0.026 rad/day and δΩ 0.045 rad/day have been found which do not confirm the solid-body hypothesis.

1.3 The Reynolds Stress

The theory of differential rotation is mainly the theory of angular momentum transport. The angular momentum equation for a turbulent rotating fluid reads as

(1.6)

Turbulence is well known to be capable of transporting momentum by the effect of turbulent viscosity. Turbulent mixing smooths out the mean velocity shear. This turbulent viscosity effect can only bring a star to the state of uniform rotation. It has been found, however, that rotating turbulence can transport angular momentum even in a state of rigid rotation. This nondiffusive transport, named the Λ effect (Rüdiger, 1989), is of key importance for understanding differential rotation of convective stars.

1.3.1 The Λ Effect

The pseudovector of angular velocity alone does not suffice to construct a polar vector of angular momentum flux. The turbulent fluid, therefore, has to possess a preferred direction for the Λ effect to emerge (Lebedinskii, 1941; Biermann, 1951). The preferred direction in stellar convection zones is provided by gravity.

The physical origin of the Λ effect is illustrated by Figure 1.12. The dashed arrows show the original motions and the solid arrows show the motions perturbed by the Coriolis force. A fluid particle, which moves originally in radius, attains azimuthal velocity, which can be estimated as , where τcorr is the characteristic time of turbulent mixing. The product is negative independently of whether the original radial motion is upward or downward. For an original azimuthal motion, radial velocity is produced by the Coriolis force, and the product is positive. On average, we have

The above expressions for the cross-correlation involve the Coriolis number

(1.7)

Figure 1.12 Illustration of angular momentum transport by rotating turbulence. The direction of rotation is shown at the top. See text.

1.3.1.1 Numerical Simulations

Käpylä and Brandenburg (2008) simulated with the Pencil Code homogeneous but anisotropic turbulence by use of variable anisotropic forcing functions in boxes with 2563 grid points. In all simulations the radial velocity fluctuations dominated the other components so that always AV < 0. Without rotation both the horizontal velocity intensities are equal. The global rotation suppresses the vertical turbulence and increases the horizontal rms values. By this influence the (negative) AV is reduced and a small but positive AH results so that a negative radial angular momentum flux and a positive horizontal angular momentum flux can be expected. Because of the rotational isotropizing of the turbulence a sufficiently rapid rotation should remarkably quench the two components of the angular momentum flux (radial and latitudinal). Figure 1.13a shows the calculated off-diagonal elements of the Reynolds stress tensor. Indeed, the resulting Qrϕ is negative while the resulting Qθϕ is positive (but small). That the radial transport of the angular momentum is indeed due to the radial anisotropy of the turbulence can be demonstrated with Figure 1.13b. There the ratio Qrϕ/AV is shown as almost independent of the numerical value of the turbulence field considered.

Figure 1.13 The Λ effect of turbulence due to anisotropic forcing. The numerical values are also the cross-correlation coefficients. (a,c) The off-diagonal elements of the normalized Reynolds tensor Qij/ vs. the colatitude. (a,b) Qθϕ and (c,d) Qrϕ (b,d) The same but for Qθϕ/(Av) and Qrϕ/(Av). The Coriolis number is fixed. From Käpylä and Brandenburg (2008). Reproduced with permission © ESO.

The anisotropy in solar and stellar convection zones can be studied in detail with 3D simulations of thermal convection (Pulkkinen et al., 1993; Chan, 2001; Käpylä, Korpi, and Tuominen, 2004; Rüdiger, Egorov, and Ziegler, 2005). In the following representation of numerical simulations with the NIRVANA CODE the Λ effect is renormalized in accordance to

(1.8)

so that

(1.9)

The anisotropy between vertical and azimuthal turbulence intensities is shown in Figure 1.15a. Without rotation the turbulence is vertically dominated except in the top layer. This is also true for the lower overshoot zone between the unstable and the stable layer (‘tachocline’). We therefore expect the occurrence of negative ΛV in the bulk of the convection zone.

The amplitude of Qθϕ is greater than the amplitude of Qrϕ, and is positive in the upper half of the convection zone. The amplitude of H, therefore, exceeds the amplitude of H by a factor of about 10 – similar to Chan’s results obtained with a completely different code. In the lower half of the convective domain it is much smaller and negative and also highly concentrated at the equator.

Motivated by this problem, and in order to avoid possible numerical artifacts of box simulations, Käpylä et al. (2011) designed global simulations in a ‘wedge’ geometry defined by 0.65 ≤ x ≤ 1, 15° ≤ θ ≤ 165° and 0 ≤ ϕ ≤ 90°. For the thermal stratification a piecewise polytrophic setup is used with the logarithmic temperature gradient which describes an unstable domain for n < 1.5. This convection zone has been sandwiched by two stable overshoot regions. The radial and latitudinal boundaries are taken to be impenetrable and stress-free, and the heat-fluxes are suppressed through the latitudinal boundaries. The simulations were performed with the PENCIL CODE code in spherical coordinates (for details see Mitra et al., 2009).

The results have been obtained with fixed low Mach number but with a free Coriolis number of order unity. The latter is insofar important as the wanted off diagonal elements of the Reynolds stress tensor only exist for sufficiently rapid rotation. Figure 1.16 presents the resulting radial flux of angular momentum Qrϕ nor malized with the turbulence intensity . Indeed, there is almost no signal for the two lowest Ω* (Figure 1.16a,b). For the intermediate rotation rates symmetric profiles with respect to the equator appear with predominantly negative signs confirming the basic result of the above presented box simulations. For the fastest rotation with Ω* 6, Qrϕ becomes smaller and even positive.

As it should, the signals for the latitudinal flux of angular momentum are antisymmetric with respect to the equator with positive values at the northern hemisphere so that the angular momentum is transported from the poles to the equator (Figure 1.17). For the fastest rotation with Ω* 6 the cross-correlation values at the top and bottom of the convection zone are rotationally quenched. The striking maxima very close to the equator which are characteristic for the box simulations (see also Hupfer, Käpylä, and Stix, 2005, 2006) no longer appear in the global simulations.

1.3.1.2 Quasi-linearTheoryof the Λ Effect

An analytical theory of the Λ effect for density-stratified convection zones was performed by Kitchatinov and Rüdiger (1993, 2005). The derivations show that the nondiffusive part, , of the velocity correlation tensor has the structure

(1.10)

Figure 1.17 The same as in Figure 1.16 but for the horizontal flux of angular momentum Qθϕ which for not too slow rotation is directed to the equator (a–f). Reproduced with permission © ESO.

The stratification in convection zones is known to be close to adiabatic. Anisotropy of stellar convection is less certain though. In the case of rapid rotation (Ω* 1), the contribution of stratification dominates and there is no arbitrariness in specifying the Λ effect. Figure 1.14, however, shows that near-surface layers are in a state of slow rotation with Ω* < 1. The resulting uncertainty in the Λ effect can be excluded by using the anisotropy resulting in 3D numerical simulations. After this is done, the Λ functions V and H assume the form shown in Figure 1.18.

Figure 1.18 The functions V(Ω*) and H(Ω*) of the Λ effect of (1.10) and (1.11) after Kitchatinov and Rüdiger (2005).

The cross-correlations

(1.11)

of the fluctuating velocities in spherical coordinates (r, θ, ϕ) are proportional to the angular momentum fluxes in radius (Qrϕ) and co-latitude (Qθϕ). The V(Ω*) function contributes to the radial flux of (1.11) only. Therefore, the part of (1.10), which includes V, represents radial flux of angular momentum. It can be shown that the angular momentum fluxes of (1.10) and (1.11) are the superposition of two nonorthogonal fluxes: V(Ω*) is the normalized flux in radius, and –H(Ω*) cos θ is normalized angular momentum flux along the rotation axis.

In the case of small Ω*, the (negative) vertical flux dominates, |V| H, and the Λ effect transports angular momentum downward. This is the reason for the existence of the radial near-surface shear seen in helioseismological inversions of Figure 1.1. In the opposite case of rapid rotation, Ω* 1, the vertical flux is small, H |V|, and the angular momentum is transported parallel to the rotation axis towards the equatorial plane. This picture basically agrees with the 3D numerical simulations presented above.

1.3.2 Eddy Viscosities

The correlation tensor of fluctuating velocities also includes the viscous part, , in line with the Λ effect2)

(1.12)

where Nijkl is the eddy viscosity tensor. The reason to write the viscosity as a tensor is the effect of rotation. Turbulent mixing becomes anisotropic under influence of rotation so that the relevant eddy viscosity depends on the orientation of the mean velocity and of the direction in which the velocity varies, relative to the rotation axis. Quasi-linear theory of turbulent transport provides the following expression for the viscosity tensor for rotating fluids

(1.13)

Five coefficients of the viscosity tensor (1.13) depend on the rotation rate,

(1.14)

The coefficient vT of (1.14) (same as in (1.10) and (1.11) for the Λ effect) is the isotropic turbulent viscosity for a nonrotating fluid, that is, this is the viscosity which would take place under actual sources of turbulence, but if there were no rotation. This coefficient, therefore, is not measurable. The mixing-length expression for nonrotating fluids,

(1.15)

Figure 1.19 Quenching functions of the eddy viscosities of (1.14) after quasi-linear theory of turbulent transport in rotating fluids (Kitchatinov, Pipin, and Rüdiger, 1994).

Equations (1.12) and (1.13) lead to the following expressions for the viscous fluxes of angular momentum

(1.16)

The first terms in the RHS of these equations are the same as in the case of isotropic viscosity. The expressions in brackets are proportional to the angular velocity gradient along the rotation axis. Therefore, v2 is the viscosity excess for the direction along the rotation axis compared to the viscosity v1, which applies to the direction normal to this axis.

The viscous fluxes of (1.16) are proportional to spatial derivatives of angular velocity. These fluxes increase with inhomogeneity of rotation. The Λ effect of (1.11), which produces this inhomogeneity, depends on rotation rate but not on the rotational shear. A steady state of differential rotation can to some extent be understood as a balance between the Λ effect and eddy viscosities. This picture is, however, very approximate and not complete because it does not allow for the angular momentum transport by the meridional flow.

1.4 The Meridional Flow

To understand the differential rotation induced by the meridional flow alone, we consider a simplified problem where the meridional flow is supposed as given, the eddy viscosity vT is isotropic, and the Λ effect in the turbulence-induced Reynolds stress is ignored (see Balbus, Latter, and Weiss, 2012). In this case, the stationary equation for the angular momentum (1.6) reduces to

(1.17)

As a further simplification, we assume a one-cell meridional flow as sketched in Figure 1.20.

It might be expected that conservation of angular momentum will result in an ‘antisolar’ rotation with angular velocity increasing with latitude. This is the case, however, only for large Reynolds number Re 1. According to Kippenhahn (1963), Steenbeck and Krause (1965) and Köhler (1969) a slow circulation with equatorward flow at the top of the convection zone can produce equatorial acceleration. Hence, any theory of the solar rotation law which only works with meridional flow as the nondiffusive angular momentum transporter leads to a circulation with downflows in the equatorial region. If large-scale upflows are observed there the equatorial acceleration must have a different origin. Note, however, that antisolar rotation can also be produced by use of a magnetic setup (see Kitchatinov and Rüdiger, 2004, and references therein).

This result for small Reynolds numbers can be understood as follows. In a steady state the material flux through the conical surface (ri – re) of constant θ in Figure 1.20 is zero, that is

(1.18)

Figure 1.20 Streamlines of meridional flow in a spherical layer with inner radius rin and outer radius re. The term rs is the radius of the stagnation point where the meridional velocity changes its sign. For low Reynolds number the direction of angular momentum transport by the meridional flow coincides with the direction of the surface flow (see text).

The angular momentum flux, tθ, produced by the meridional flow, through the same surface reads as

(1.19)

Differential rotation in the case of small Reynolds number is also small. Therefore, we can put Ω in (1.19) outside of the integral sign. On doing so and using (1.18), the angular momentum flux (1.19) can be written as

(1.20)

The situation is different for large Reynolds numbers, Re 1. Then the second term in the RHS of (1.17) can be neglected to write this equation as

(1.21)

where ψ is the stream function of the meridional flow

(1.22)

The change of character of the differential rotation with Reynolds number is shown in Figure 1.21. The latitudinal differential rotation depends on the sense of meridional flow for Re 1, while a fast flow always produces antisolar rotation for Re 1 independent of its orientation. The transition from slow to fast circulation regimes occurs at Reynolds numbers of about 100. The Sun with meridional flow of order 10m/s and eddy viscosity vT 5 × 1012cm2/s belongs to the case of moderate Reynolds numbers, Re ~ 10.

Figure 1.21 Latitudinal differential rotation produced by the meridional flow as a function of the Reynolds number. Clockwise flow (solid), counterclockwise flow (dashed). Without Λ effect positive k are only possible for clockwise flow.

If the helioseismology finds for the solar meridional circulation a one-cell counterclockwise pattern which rises (sinks) in the equatorial (polar) region then the observed equatorial acceleration must be a result of the angular momentum transport by Reynolds stress (in form of the Λ effect) – and cannot be a result of the angular momentum transport of the (say) baroclinic flow. With the empirical findings of Section 1.1.3 concerning the meridional circulation geometry it is not possible to explain the solar rotation law as due to the action of baroclinic flows.

1.4.1 Origin of the Meridional Flow

Centrifugal and buoyancy forces are the most essential drivers of meridional flow in stellar convection zones. The buoyancy force results from a slight dependence of mean temperature on latitude. Both forces are strong: each of them alone would drive a meridional flow of several hundreds of meters per second on the Sun (Durney, 1996). The forces, however, almost balance each other and the meridional flow results from the slight imbalance between them.

The meridional flow equation can be obtained as azimuthal component of the ‘curled’ (1.6). The resulting steady equation reads as

(1.23)

(1.24)

where the first subscript on the right-hand side indicates the azimuthal component in spherical coordinates.

(1.25)

Such a large difference in characteristic values of the left and right-hand sides of (1.23) shows that this equation might be satisfied by the two terms of the RHS balancing each other. This state is known as the ‘thermal wind balance.’ Meridional flow arises from slight deviations from the balance. It is remarkable that the flow in turn maintains the balance. Any considerable deviation from the balance would produce a large meridional flow which reacts back on the distributions of angular velocity and entropy to re-establish the balance. A successful theory must, therefore, keep the meridional flow in both the angular momentum and entropy equations. A ‘self-influence’ of the meridional flow is, however, of secondary importance and the term nonlinear in Um is dropped in (1.23). It is small compared to the term nonlinear in the angular velocity.

If angular velocity decreases with distance from the equatorial plane (dΩ/dz 0), the (nonconservative) centrifugal force is largest near the equator. The resulting torque drives meridional circulation with poleward flow at the top of the convection zone. From the early model of Köhler (1969, 1970) to the formulations of Glatzmaier (1985), Gilman and Miller (1986), Brandenburg et al. (1990), Miesch et al. (2000) and Brun and Toomre (2002), the inclusion of a meridional flow always led to the ‘Taylor-number puzzle’: cylindrical isorotation contours are found, independently of the Λ effect applied. The isolines of Ω in the solar convection zone, however, are not cylindrical (Figure 1.1). The above estimations show that positive differential temperature with the poles warmer than the equator by about 1 K suffices to produce a considerable deviation from the cylindrical rotation law.

The impact on the maintenance of differential rotation can be seen from the equation for the meridional flow. For fast rotation (large Taylor number) the Reynolds stress and nonlinear terms can be neglected and (1.23) is reduced to

(1.26)

It follows that a gradient of the angular velocity along the axis of rotation is needed to balance a horizontal temperature gradient. A disk-shaped rotation pattern, therefore, results in the polar area (only) due to the action of the baroclinic flow. For warmer poles (δT sinks equatorwards) the z-gradient of Ω must be negative as observed. The meridional flow without the baroclinic component only yields dΩ/dz 0 (see Figure 1.32). Hence, the empirical finding of negative dΩ/dz at the northern polar axis proves the existence of baroclinic flows in the solar convection zone.

As we shall demonstrate by means of Figure 1.30, the superrotation beneath the solar equator is a direct indication for the action of the Λ effect in the convection zone. All models without Λ effect but with a clockwise (equatorward at the surface) circulation lead to dΩ/dr 0 in the bulk of the convection zone beneath close to the equator. Such clockwise flows are able to accelerate the equator relative to the poles but they cannot produce the superrotation beneath the equator.

1.4.2 The Differential Temperature

Heat transport in a stellar convection zone is governed by the mean entropy equation,

(1.27)

where ε is the source function, Frad and Fconv are the radiative and convective heatfluxes3).

The differential temperature is currently understood as an outcome of rotational influence on convective heat transport. The tensor χij of the eddy thermal diffusion that controls the convective heat-flux,

(1.28)

includes the rotationally induced anisotropy and quenching

(1.29)

The diffusivity quenching functions are shown in Figure 1.22, and the background diffusivity χT can be expressed in terms of the entropy gradient,

(1.30)

similar to the (1.15) for the eddy viscosity.

Figure 1.22 Quenching functions of the eddy thermal diffusivity of (1.29) after quasi-linear theory of turbulent transport in rotating fluids (Kitchatinov, Pipin, and Rüdiger, 1994).

Weiss (1965) suggested that the eddy thermal diffusivity in rotating fluids depends on latitude. Equation (1.30) allows for the dependence:

(1.31)

The diffusivity increases with latitude to imply that the poles are warmer than the equator. A differential temperature of this sense is required to explain the equatorial acceleration of Figure 1.1 (Gilman, 1986; Durney, 1987). However, the latitude-dependent heat transport does not help to resolve the Taylor-number puzzle. Much more significant is the anisotropy of the eddy heat transport. The diffusivity tensor (1.29) possesses the finite off-diagonal component

(1.32)

This means that even if the mean entropy varies only with radius, the convective heat-flux deviates from radial direction to poles, Fθconv cos θ < 0. The differential temperature resulting from the convective heat-flux in latitude is sufficiently large (~ 1 K) to resolve the Taylor-number puzzle.

There were many attempts to observe the differential temperature on the Sun. Recent observations of Rast, Ortiz, and Meisner (2008) suggest that the differential temperature on the solar photosphere is indeed positive, δT 2.5 K.

We always find a higher temperature at high latitudes than at the equator. This is a consequence of the tilt of the convective heat transport vector towards the axis of rotation caused by the Coriolis force. In spherical polar coordinates the components of the heat-flux read as

(1.33)

(1.34)

The first term in the horizontal component precludes a purely radial stratification. Any variation of the specific entropy with the radius will cause a horizontal heat-flux and thus build up a horizontal gradient. This case is profoundly different from that of a latitude-dependent but still purely radial heat-flux. The latter would have a much smaller impact and would result in a much weaker differential rotation.

Figure 1.23a shows the depth profile of the correlation (ur) with the temperature fluctuation for various latitudes which is positive (negative) in the convection zone (overshoot layer). More important is that the values differ between poles and equator. The pole–equator difference in the radial heat-flux slightly depends on the radius. Except for the top layer, the eddy heat-flux at the equator exceeds the eddy heat-flux at the poles. In the top layer, however, where the turbulence is horizontally dominated, the polar heat-flux dominates the equatorial one. Rieutord et al.