Introduction to Hydromagnetic Dynamo Theory with Applications to the Sun and the Earth

Home , Dynamo theory

Introduction Basic electrodynamics Kinematic turbulent dynamos The solar dynamo Magnetohydrodynamical dynamos and geodynamo simulations

Dieter Schmitt

Max Planck Institute for Solar System Research

IMPRS Solar System School 18–20 June 2013

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos The solar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Outline Mean-field coefficients 1 Introduction Mean-field dynamos Magnetic field of the Earth 4 The solar dynamo Magnetic field of the Sun Basic incredients Dynamo hypothesis Convection zone dynamos Homopolar dynamo Overshoot layer dynamos 2 Basic electrodynamics Interface dynamos Pre-Maxwell theory Flux transport dynamos Induction equation 5 Magnetohydrodynamical dynamos Alfven’s theorem and geodynamo simulations Magnetic Reynolds number Equations and parameters Poloidal and toroidal magnetic Proudman-Taylor theorem and fields convection in a rotating sphere 3 Kinematic turbulent dynamos A simple geodynamo model Antidynamo theorems Advanced models Parker’s helical convection Reversals Mean-field theory 6 Literature

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Magnetic field of the Earth Basic electrodynamics Magnetic field of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Geomagnetic field

1600 Gilbert, De Magnete: ”Magnus magnes ipse est globus terrestris.” (The Earth’s globe itself is a great magnet.)

1838 Gauss: Mathematical description of geomagnetic ﬁeld l+1 P m − P ∇ m − P ∇ R m m m B = l,m Bl = Φl = R r Pl (cos ϑ) gl cos mφ + hl sin mφ sources inside Earth l number of nodal lines, m number of azimuthal nodal lines l = 1, 2, 3,... dipole, quadrupole, octupole, ... m = 0 axisymmetry, m = 1, 2,... non-axisymmetry 0 ≈ − | m| | m| Earth: g1 0.3 G, all other gl , hl < 0.05 G h i1/2 3 0 2 1 2 1 2 ≈ · 25 3 mainly dipolar, dipole moment µ = R (g1) + (g1) + (h1 ) 8 10 G cm h i1/2 . 1 2 1 2 0 ≈ ◦ tan ψ = (g1) + (h1 ) g1, dipole tilt ψ 11 dipole : quadrupole ≈ 1 : 0.14 (at CMB)

CMB

ICB

Br at surface 1990

= l

Br at CMB 1990

Br at CMB 1890

westward drift 0.18◦/yr u ≈ 0.5 mm/sec

Br at CMB 1990

SINT-800 VADM (Guyodo and Valet 1999)

NGP (Ohno and Hamano 1992)

(Hale 1908)

60 Zyklus 19 Zyklus 21

40 O

-20 Heliographische Breite [ ] -40 Zyklus 18 Zyklus 20 -60 1950 1960 1970 1980 1990 Jahr Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Magnetic field of the Earth Basic electrodynamics Magnetic field of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Polar fields and cycle predictions

268 CAMERON ET AL. Vol. 719 dynamos, the correlation constrains the subsurface dynamics (see, for example, Yeates et al. 2008).

5. PARAMETER DEPENDENCE In the previous section, we showed that a good representation of the empirically determined open flux Fopen could be achieved with the parameter values of the reference model. Figure 8 shows the effect of separately varying the initial field strength, B0,the surface diffusivity, ηH , the radial diffusivity, ηr , and the cusp surface height, Rcusp. In all panels except the lower right we have kept Bmax constant: in this panel we recalibrated Bmax to account for the change in the total unsigned flux resulting from the change in ηr . The upper left panel of Figure 8 shows the effect of changing B0. Since B0 describes the initial polar field, varying this value leads to an offset in the strength of the axial dipole moment, which persists throughout the simulation when ηr = 0. This (Schattenoffset results in alternating cycles having either stronger or weaker axial dipole moments depending on whether or not they ethave al. the same 1978) sign of dipole moment as that of the initial state. Near activity minima it is the lower order axial moments that dominate Fopen so that the minima alternately become higher (Cameronand lower. The upper right panel of Figure 8 shows the effect of etincreasing al. 2010)ηH : the minima of Fopen are shifted upward, while the maxima are not substantially affected. The explanation for the upward shift is that ηH determines the amount of flux that (Wangcrosses the and equator and thus directly influences the axial dipole moment. There is also a weak but noticeable 22 year component, Sheeleywith the minima 2009) of alternating cycles being weaker. This 22 year component is present because we have not recalibrated B0. Figure 7. Correlation between the observed maximum polar field from the flux The middle left panel of Figure 8 shows the effect of transport model and the preceding (upper panel) and subsequent (lower panel)Dieter Schmitt Hydromagnetic Dynamo Theory cycle’s maximum sunspot area. The asterisks represent the values for individual increasing ηr . The enhanced decay of the field reduces Fopen cycles: the solid line is a linear fit. The maximum sunspot area is uncorrelated not only around the minima but also during the rising phase of (r =−0.1) with the subsequent polar field maximum, but strongly correlated a cycle. This leads to too low minima and a delay of the rising (r = 0.85) with the preceding polar field maximum. The first maximum of phase. There is again a 22 year component because B0 has not the polar field has been omitted from the analysis since it is affected by the arbitrariness of the initial field. been recalibrated. The middle right panel shows the effect of varying the tilt angle reduction factor, g,from0.7 to 1, which modifies the Figure 6 presents the time evolution of the unsigned north–south magnitude of the polar fields and the axial dipole moment. The averaged polar field from the simulation as well as the observed signature is therefore an increase in the magnitude of the changes strength of the solar activity. Ignoring the first cycle, which is in the dipole moment (thus Fopen) and its minima, so that the strongly affected by the initial condition, the polar field can be effect almost cancels after two cycles. This also produces a seen to anticipate the peaks of solar activity. We have quantified strong 22 year periodicity in the minima. We comment that this relationship by calculating the correlation between the g = 0.7 is required to obtain the correct ratio between the strength of peaks of the (unsigned north–south averaged) polar maxima and minima of the open flux, as it essentially scales the field and the strength of the adjacent activity maxima. As can be low-order axial multipoles while barely affecting the equatorial seen in Figure 7, the maxima of the polar field between cycles multipoles. Introducing g does not affect whether or not the n and n + 1 reflect the amplitude of cycle n + 1 much better than polar fields reverse—the 22 year periodicity, when g is varied cycle n. The correlation coefficient for the relationship between in isolation, can be removed by an appropriate choice of B0. the polar field and the amplitude of the next cycle is 0.85. This is The lower left panel shows that increasing Rcusp in isolation in contrast to the result of Cameron & Schussler¨ (2007) who did weakens Fopen. The effect is strongest during the maxima as not consider a cycle-to-cycle-dependent tilt angle; they found it preferentially reduces the contribution from higher order that the polar fields and strength of the polar field closely follow multipoles. The influence is thus qualitatively different from the previous cycle. This indicates the importance of the cycle- that of the other parameters in that it changes the relative to-cycle variations in the tilt angle. contributions of the different multipoles. Some observational evidence concerning the correlation be- In the panels discussed so far, we have kept Bmax, the scaling tween the polar field and the strength of the previous and subse- factor for the total flux of newly emerging BMRs, constant. quent cycles has been previously considered, e.g., by Schatten Varying ηr as was done in the middle left panel changes the et al. (1978), Layden et al. (1991), Svalgaard et al. (2005), and total amount of unsigned flux, and so affects the calibration of Jiang et al. (2007). The existence of a correlation between the Bmax. In the bottom right panel, we therefore show the effect of polar field and the strength of the next cycle is evidence in favor a change in ηr together with the corresponding change in Bmax. of a Bacock-Leighton-type dynamo. Within the context of such Since the entire system is linear in Bmax; changing Bmax merely Introduction Magnetic field of the Earth Basic electrodynamics Magnetic field of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Variability of cycle length and strength

200 19 Dalton 18 21 150 Maunder Minimum Minimum 17 20 100 16

50 Sunspot number 0 1600 1700 1800 1900 2000 Year Waldmeier (1935) rule

odd-even effect (Gnevyshev and Ohl 1948)

Dieter Schmitt Hydromagnetic Dynamo Theory

FIG. 1.ÈBrief look at solar cycle variability; Fig. 1a is a time series of the smoothed monthly sunspot number, and Fig. 1b the corresponding power spectrum. Fig. 1c illustrates the anticorrelation existing between the amplitude of the sunspot cycle, (here measured by the peak smoothed monthly sunspot number) and its duration. This anticorrelation is statistically signiÐcant (linear correlation coefficient r \[0.37, null hypothesis rejected with probability 0.91). Fig. 1d shows the observed variations in the sunspot cycle period, from 1600 to the present. Cycle periods prior to 1750 (diamonds) are based on 10Be ice core data, and taken directly from Beer et al. (1998), with error bars estimated by these authors to be in the range ^0.5È1yr(^0.75 yr is used here). Later cycle periods ( Ðlled circles) are estimated from the smoothed monthly sunspot numbers, and are accurate to ^1 month. The straight lines are linear least-squares Ðts to the variation with time of cycle duration, for the sunspot data only (dotted line), and for the combined 10Be]sunspot data (dashed line). In the latter case the mean period is 10.83 yr. Fig. 1e shows the cumulative ““ phase error ÏÏ between the actual dates of minima and those computed from the weak long term trends corresponding to the least-squares Ðts in Fig. 1d. Here as in later Ðgures of this type, a phase error of n corresponds to a full sunspot cycle, i.e., half a magnetic cycle. The cumulative errors are calculated starting either from the year 1594 (diamonds) or from sunspot cycle one ( Ðlled circles). Introduction Magnetic ﬁeld of the Earth Basic electrodynamics Magnetic ﬁeld of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Long-term variability / C14 and Be10 LONG-TERM INDIRECT INDICES OF SOLAR VARIABILITY 59 Maunder Spörer20 (Stuiver et al. 1998) 10 Dalton 150 0

-10 100

-20

14 C 50 -30 [permil] 1500 1600 1700 1800 1900 2000

0 LONG-TERM INDIRECT INDICES OF SOLAR VARIABILITY 61

1.8 -50 a Dye 3

8000 6000 4000 2000 0 2000 ] 1.6 -1

Year [BC / AD] g 1.4 Figure 5. 114C measured in tree rings for the past 11 ky. The record is characterised1.2 by short-term atoms variations superimposed to a long-term trend. The short-term variations can4 be attributed1 mainly to solar modulation whereas the long-term trend is mainly the result of variations of0.8 the geomagnetic Be [10

ﬁeld intensity during the past 20 ky and possibly some changes within the carbon10 0.6 system. The period from 1510 AD to 1950 AD is shown in detail. The peak at 1700 AD0.4 corresponds to the maximum of the Maunder minimum, a period famous for its almost complete lack0.2 of sunspots. The 14 steep decrease during the 20th century is due(Beer to the etSuess al. effect 1994) (dilution of the atmospheric1440 1520C 1600 1680 1760 1840 1920 Year AD content by the growing combustion of fossil fuel) and an increase in solar activity.60

Dieter Schmitt ] Hydromagneticb Dynamo Theory

-1 GISP II y 50 -2 solar activity occurred around 1800 (Dalton Minimum) and 1900. The decreasing trend in 114C as from about 1900, however, is to a large extent of40 anthropogenic 5.5 10 atoms cm origin. The increasing rate of fossil fuel consumption led to an increasing4 amount 14 14 12 30 of C-free CO2 reducing the C/ C ratio considerably.

14 Be [10

The 1 C record covering the past 11500 years (Fig. 5) is based10 20 on tree rings. Comparison of the ring width between different trees enabled to match the individual records to a continuous chronology. Before 11500 BP, during theYear deglaciation, BP 2.5 10 4 2 10 4 1.5 10 4 1 10 4 5000 80

the vegetation was rather different and the number of trees much sparser.] c Therefore,

-1 70 GRIP no continuous record is available yet although several ﬂoating y chronologies are under construction. Instead of tree rings, other material such as-2 varved60 sediments (Hajdas, 1993; Hughen et al., 1998; Kitagawa and Van der Plicht,50 1998), corals 40 (Bard et al., 1990), or stalagmites have been used. The absolute atoms cm age can be deter- 4 mined by counting the annual sediment layers or using the U/Th dating30 technique.

However, all these techniques lead to somewhat larger uncertaintiesBe [10 20 in 114C due to 10 14 dating difficulties or the fact that they do not record directly the atmospheric10 1 C (corals, stalagmites). Year BP 4 5 104 4.5 104 4 104 3.5 104 3 104 2.5 104 Since solar variability is at least partly quasi-periodicFigure 6. 10Be itrecords is interesting with different to search time resolutions for different time intervals. for periodicities. The most famous periodicitya) Annual of solar10Be variability concentrations is the from 11-year Dye 3 after removing the short-term fluctuations. b) 10Be flux Schwabe cycle. Although this periodicity variescovering considerably the transition between from the 7 last and Glaciation 18 into the Holocene . The sharp peaks correspond years since 1750 it seems quite stable and closeprobably to 11 to years quiet if Sun averaged periods (e.g.over Maunder longer Minimum). time periods. Due to the dampening effect ofc) the10 carbonBe flux duringcycle system, Marine Isotope the Schwabe Stage 3 mainly. The peak close to 40 ky BP and the long-term cycle is attenuated strongly and difficult to detectchanges in the are1 consistent14C (Fig. with 5) changes(Siegenthaler of the global mean production rate induced by the geomagnetic field intensity.

with the density ρ (g cm−3), the accumulation rate a (cm y−1), and the concentration c (104 atoms/g). Since ρ of ice is constant, the flux F is mainly determined by the product of concentration and accumulation rate. During the Holocene the accumulation rate is relatively constant leading to a flux F that looks much the same as the concentration c. During glacial times, however, the accumulation rate changes by up to a factor of 2. As has been shown in earlier studies the 10Be flux can be considered as proportional to the mean global production rate (Wagner et al., 2000a). As can be seen in Fig. 6 the 10Be records vary on very different time scales. The short-term variations of the 10Be concentration in Fig. 6a reflect the 11-year Introduction Magnetic field of the Earth Basic electrodynamics Magnetic field of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Solar activity in the last 11400 years

(Solanki et al. 2004)

Dieter Schmitt Hydromagnetic Dynamo Theory 7

2.4 Faraday rotation and Faraday depolarization

The linearly polarized radio wave is rotated by the Faraday effect in the passage through a magneto- ionic medium (Fig. 1). This effect gives us another method of studying magnetic fields – their regular component along the line of sight. The rotation angle Ф induced in a polarized radio wave is given by: G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 95 2 Ф = k λ ∫ ne B║ dl The magnetic field is produced by convectively-driven dynamo detecting short timescale secular variation, and constraining the action in the highly electrically conducting metallic hydrogen planet’s internal structure (Connerney, private communication). with λ wavelength of observation n thermal electron density mantle and in the less electrically conducting region near the base e B║ strength of the regular magnetic field component along the line of sight of the molecular envelope (Stanley and Glatzmaier, 2010). The ex- 2.1.5. Saturn dl pathlength along the magnetic field act location and extent of the dynamo region is poorly constrained. Saturn was revealed to have a magnetic field during the Pioneer Nellis (2000) argues that the dynamo can extend up to 95% of the 11 flyby in 1979 (Smith et al., 1980), and subsequent observations and k is a constant (see below). In practice, the parameter Faraday Depth (FD) is used (Burn 1966): planet radius. Consequently, the dynamo region is expected to ex- were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 2 hibit significant changes in density and electrical conductivity, the in 1981 (Ness et al., 1982). The Cassini spacecraft arrived at Saturn Ф = FD λ

effects of which are relatively unknown. The upcoming Juno mis- in 2004 and is still collecting data (Dougherty et al., 2005; Burton -2 where FD = 0.81 L (rad m ) sion will critically improve our understanding of Jupiter by resolv- et al., 2009). These observations have mapped the large-scale radial -3 ing the magnetic field up toG. spherical Schubert, K.M. harmonic Soderlund / degree Physics of 14,the Earthmagnetic and Planetary field, Interiors shown 187 (2011) in Fig. 92–108 2d, and suggest that no resolved95 with ne thermal electron density in cm B║ regular field strength in µGauss L pathlength in parsec. The magnetic field is produced by convectively-driven dynamo detecting short timescale secular variation, and constraining the G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 95 action in the highly electrically conducting metallic hydrogen planet’s internal structureIntroduction (Connerney, private communication). The observable quantity Rotation Measure (RM = ΔФ / Δλ2) is identical to the physical quantity FD only Magnetic field of the Earth 2 The magnetic field is produced by convectively-driven dynamomantle anddetecting in the less short electrically timescale(a) conductingMercury secular variation, region near and the constraining base the Basic(b) electrodynamics Ganymede in the rare cases when Ф is a linear function of λ . If the rotating region is located in front of the action in the highly electrically conducting metallic hydrogenof the molecularplanet’s envelope internal ( structureStanley and (Connerney, Glatzmaier, private 2010). communication). The ex- 2.1.5. Saturn Magnetic field of theemitting Sun region (“Faraday screen”), RM = FD. In case of a single emitting and rotating region with a act location and extent of the dynamo region is poorly constrained. KinematicSaturn was revealed turbulent to have dynamos a magnetic field during the Pioneer symmetric magnetic field profile, RM ≈ FD / 2 if Faraday depolarization (see below) is small. mantle and in the less electrically conducting region near the base Dynamo hypothesis of the molecular envelope (Stanley and Glatzmaier, 2010). TheNellis ex- (2000)2.1.5.argues Saturn that the dynamo can extend up to 95% of the 11 flyby in 1979The (Smith solar et al., dynamo 1980), and subsequent observations As Faraday rotation angle is sensitive to the sign of the field direction, only regular fields give rise to act location and extent of the dynamo region is poorly constrained.planet radius.Saturn Consequently, was revealed the dynamo to have region a magnetic is expected field during to ex- the Pioneerwere made by Voyager 1 in 1980 (Ness et al., 1981) andHomopolar Voyager 2 dynamoFaraday rotation, while anisotropic and turbulent fields do not. For typical plasma densities and regular Nellis (2000) argues that the dynamo can extend up to 95% ofhibit the significant11 flyby changes in 1979Magnetohydrodynamical in (densitySmith et and al., electrical 1980), and conductivity, subsequent dynamos the observationsin and 1981geodynamo (Ness et al., 1982). simulations The Cassini spacecraft arrived at Saturn field strengths in the interstellar medium of galaxies, Faraday rotation becomes significant at wavelengths larger than a few centimeters. Only in the central regions of galaxies, Faraday rotation is planet radius. Consequently, the dynamo region is expected toeffects ex- of which are relatively unknown. The upcoming Juno mis- in 2004 and is still collecting data (Dougherty et al., 2005; Burton were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 strong already at 1-3 cm wavelengths. Measurements of the Faraday rotation angle from multi- hibit significant changes in density and electrical conductivity,sion the will criticallyin 1981improve (Ness etal., our 1982 understanding). The Cassini of Jupiter spacecraft by resolv- arrived at Saturnet al., 2009). These observations have mapped the large-scale radial wavelength observations allow determination the strength and direction of the regular field component effects of which are relatively unknown. The upcoming Junoing mis- the magneticin 2004 and field is still up collecting to spherical data harmonic (Dougherty degree et al., 14,2005; Burtonmagnetic field, shown in Fig. 2d, and suggest that no resolved along the line of sight. Its combination with the total intensity and the polarization pseudo-vectors sion will critically improve our understanding of Jupiter by resolv-Otheret al., 2009). These observations cosmic have mapped the large-scale bodies radial yields in principle the three-dimensional picture of galactic magnetic fields and the three field ing the magnetic field up to spherical harmonic degree 14, magnetic field, shown in Fig. 2d, and suggest that no resolved components regular, anisotropic and turbulent.

By definition the regular magnetic fields point towards the observer when RM > 0. The quantity (a) Mercury (b) Ganymede is the average of the product (ne B║) along the line of sight which generally is not equal to the product of the averages if fluctuations in ne and B║ are correlated or anticorrelated. As a -0.6 µT0 0.6 µT -1.5 µT0 1.5 µT (a) Mercury (b) Ganymede consequence, the field strength cannot be easily determined from RM even if additional Planets Stars information about is available, e.g.Galaxies from pulsar dispersion measures (section 3.3). e

-0.6 µT0 0.6 µT -1.5 µT0 1.5 µT

-0.6 µT0 0.6 µT -1.5 µT0 1.5 µT (c) Jupiter (d) Saturn Zeeman Fig. 1: Synchrotron emission and Faraday rotation (c) Jupiter-1200 µT (d) Saturn0 1200 µT -60 µT 0 60 µT

Saturn (e) UranusCa II H+K (f) Neptune

-1200 µT0 1200 µT -60 µT 0 60 µT M31

-1200 µT0 1200 µT -60 µT (e) Uranus0 60 µT (f) Neptune (e) UranusUranus -100 µT (f) Neptune0 100 µT DI and-100 µT ZDI 0 100 µT Fig. 2. Radial magnetic ﬁeld at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3).

-100 µT0 100 µT -100 µT0 100 µT M51

-100 µT0 100 µT Fig. 2. Radial magnetic-100 field atµT the surfaces of (a)0 Mercury, (b) 100 Ganymede, µT (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). Dieter Schmitt Hydromagnetic Dynamo Theory Fig. 2. Radial magnetic field at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). Introduction Magnetic field of the Earth Basic electrodynamics Magnetic field of the Sun Kinematic turbulent dynamos Dynamo hypothesis The solar dynamo Homopolar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Dynamo hypothesis

Larmor (1919): Magnetic field of Earth and Sun maintained by self-excited dynamo Dynamo: u×B y j y B y u Faraday Ampere Lorentz motion of an electrical conductor in an ’inducing’ magnetic field y induction of electric current Self-excited dynamo: inducing magnetic field created by the electric current (Siemens 1867) Example: homopolar dynamo Homogeneous dynamo (no wires in Earth core or solar convection zone) y complex motion necessary Kinematic (u prescribed, linear) Dynamic (u determined by forces, including Lorentz force, non-linear)

u uxB

electromotive force u×B y electric current through wire loop y induced magnetic ﬁeld reinforces applied magnetic ﬁeld self-excitation if rotation Ω > 2πR/M is maintained where R resistance, M inductance

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Pre-Maxwell theory Basic electrodynamics Induction equation Kinematic turbulent dynamos Alfven’s theorem The solar dynamo Magnetic Reynolds number Magnetohydrodynamical dynamos and geodynamo simulations Poloidal and toroidal magnetic ﬁelds Pre-Maxwell theory

Maxwell equations: cgs system, vacuum, B = H, D = E ∂E ∂B c∇×B = 4πj + , c∇×E = − , ∇·B = 0 , ∇·E = 4πλ ∂t ∂t

Basic assumptions of MHD: • u c: system stationary on light travel time, no em waves • high electrical conductivity: E determined by ∂B/∂t, not by charges λ E B E 1 L u e E2 c ≈ y ≈ ≈ 1 , E plays minor role : el ≈ 1 2 L T B c T c em B ∂E/∂t E/T E u u2 ≈ ≈ ≈ 1 , displacement current negligible c∇×B cB/L B c c2

Pre-Maxwell equations: ∂B c∇×B = 4πj , c∇×E = − , ∇·B = 0 ∂t

Pre-Maxwell equations Galilei-covariant: 1 E0 = E + u×B , B0 = B , j0 = j c Relation between j and E by Galilei-covariant Ohm’s law: j0 = σE0 in resting frame of reference, σ electrical conductivity 1 j = σ(E + u×B) c

Magnetohydrokinematics: Magnetohydrodynamics: c∇×B = 4πj additionally ∂B c∇×E = − Equation of motion ∂t Equation of continuity ∇·B = 0 Equation of state 1 j = σ(E + u×B) Energy equation c Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Pre-Maxwell theory Basic electrodynamics Induction equation Kinematic turbulent dynamos Alfven’s theorem The solar dynamo Magnetic Reynolds number Magnetohydrodynamical dynamos and geodynamo simulations Poloidal and toroidal magnetic ﬁelds Induction equation

Evolution of magnetic ﬁeld ! ! ∂B j 1 c 1 = −c∇×E = −c∇× − u×B = −c∇× ∇×B − u×B ∂t σ c 4πσ c ! c2 = ∇×(u×B) − ∇× ∇×B = ∇×(u×B) − η∇×∇×B 4πσ c2 with η = = const magnetic diffusivity 4πσ induction, diffusion

∇×(u×B) = −B ∇·u + (B·∇)u − (u·∇)B

expansion/contraction, shear/stretching, advection

∇·B = 0 as initial condition, conserved

∂B Ideal conductor η = 0 : = ∇×(u×B) ∂t d Z Magnetic ﬂux through ﬂoating surface is conserved : B·dF = 0 dt F

Proof: Z Z Z Z I 0 = ∇·BdV = B·dF = B(t)·dF − B(t)·dF0 − B(t)·ds×udt , F F0 C Z Z Z I B(t +dt)·dF0 − B(t)·dF = B(t +dt) − B(t) ·dF − B·ds×udt F0 F F C ! ! Z ∂B I Z I = dt ·dF − B·ds×u = dt ∇×(u×B)·dF − B·ds×u ∂t C C I I ! = dt u×B·ds − B·ds×u = 0 C C

Frozen-in field lines impression that magnetic field follows flow, but E =−u×B/c and c∇×E =−∂B/∂t ∂B = ∇×(u×B) = −B ∇·u + (B·∇)u − (u·∇)B ∂t (i) star contraction: B ∼ R−2, ρ ∼ R−3 y B ∼ ρ2/3 Sun y white dwarf y neutron star: ρ [g cm−3]: 1 y 106 y 1015 (ii) stretching of flux tube: Bd2 = const, ld2 = const y B ∼ l (iii) shear, differential rotation

∂Bφ/∂t = r sin θ ∇Ω·Bp

Dimensionless variables: length L, velocity u0, time L/u0

∂B u L = ∇×(u×B) − R−1 ∇×∇×B with R = 0 ∂t m m η

as combined parameter

laboratorium: Rm 1, cosmos: Rm 1

induction for Rm 1, diffusion for Rm 1, e.g. for small L example: ﬂux expulsion from closed velocity ﬁelds

(Weiss 1966)

Spherical coordinates (r, ϑ, ϕ)

Axisymmetric ﬁelds: ∂/∂ϕ = 0

B(r, ϑ) = (B , B , B ) =0 r ϑ ϕ z}|{ 2 1 ∂r B 1 ∂ sin ϑB 1 ∂Bϕ ∇·B = 0 y r + ϑ + = 0 r2 ∂r r sin ϑ ∂ϑ r sin ϑ ∂ϕ

B = Bp + Bt poloidal and toroidal magnetic ﬁeld

Bt = (0, 0, Bϕ) satisﬁes ∇·Bt = 0

Bp = (Br , Bϑ, 0) = ∇×A with A = (0, 0, Aϕ) satisﬁes ∇·Bp = 0 ! 1 ∂r sin ϑAϕ ∂r sin ϑAϕ B = , − , 0 p r sin ϑ r∂ϑ ∂r axisymmetric magnetic ﬁeld determined by the two scalars: r sin ϑAϕ and Bϕ

Axisymmetric fields: c c j = ∇×B , j = ∇×B t 4π p p 4π t r sin ϑAϕ = const : field lines of poloidal field in meridional plane

ﬁeld lines of Bt are circles around symmetry axis

Non-axisymmetric ﬁelds:

B = Bp + Bt = ∇×∇×(Pr) + ∇×(Tr) = −∇×(r ×∇P) − r ×∇T r = (r, 0, 0) , P(r, ϑ, ϕ) and T(r, ϑ, ϕ) defining scalars c c ∇·B = 0 , j = ∇×B , j = ∇×B t 4π p p 4π t r ·Bt = 0 field lines of the toroidal field lie on spheres, no r component

Bp has in general all three components

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Antidynamo theorems Basic electrodynamics Parker’s helical convection Kinematic turbulent dynamos Mean-field theory The solar dynamo Mean-field coefficients Magnetohydrodynamical dynamos and geodynamo simulations Mean-field dynamos Cowling’s theorem (Cowling 1934)

Axisymmetric magnetic ﬁelds can not be maintained by a dynamo.

Sketch of proof: electric currents as sources of the magnetic field only in finite space field line F = 0 along axis closes at infinity field lines on circular tori whose cross section are the lines F = const z F = 0 z F = const

grad F = 0 or B p = 0 O-type neutral point O-type neutral line s axisymmetry: closed neutral line

around neutral line is ∇×B , 0 y jϕ , 0, but there is no source of jϕ: Eϕ = 0 because of axisymmetry and (u×B)ϕ = 0 on neutral line for ﬁnite u

Consider vicinity of neutral line, assume axisymmetry I I Z 4π Z 4π Z B dl = B·dl = ∇×B df = j·df = |j |df p c c ϕ 4πσ Z 4πσ Z 4πσ Z = |up ×Bp|df ≤ upBpdf ≤ up,max Bpdf c2 c2 c2 integration circle of radius ε

4πσ 2 2πσ Bp2πε ≤ up,maxBpπε or 1 ≤ up,maxε c2 c2 ε → 0 y up,max → ∞ contradiction

Toroidal velocity theorem (Elsasser 1947, Bullard & Gellman 1954) A toroidal motion in a spherical conductor can not maintain a magnetic ﬁeld by dynamo action.

Sketch of proof: d (r ·B) = η∇2(r ·B) for r ·u = 0 dt y r ·B → 0 for t → ∞ y P → 0 y T → 0

Toroidal ﬁeld theorem / Invisible dynamo theorem (Kaiser et al. 1994) A purely toroidal magnetic ﬁeld can not be maintained by a dynamo.

velocity u

vorticity ω = ∇×u

helicity H = u·ω

(Parker 1955)

Statistical consideration of turbulent helical convection on mean magnetic ﬁeld (Steenbeck, Krause and Rädler 1966) ∂B = ∇×(u×B) − η∇×∇×B ∂t u = u + u0 , B = B + B0 Reynolds rules for averages ∂B = ∇×(u×B + E) − η∇×∇×B ∂t E = u0 ×B0 mean electromotive force ∂B0 = ∇×(u×B0 + u0 ×B + G) − η∇×∇×B0 ∂t G = u0 ×B0 − u0 ×B0 usually neglected , FOSA = SOCA B0 linear, homogeneous functional of B approximation of scale separation: B0 depends on B only in small surrounding 0 0 Taylor expansion: u ×B = αijBj + βijk ∂Bk /∂xj + ... i

0 0 u ×B = αijBj + βijk ∂Bk /∂xj + ... i 0 αij and βijk depend on u 0 homogeneous, isotropic u : αij = αδij , βijk = −βεijk then u0 ×B0 = αB − β∇×B Ohm0s law: j = σ(E + (u×B)/c) j = σ(E + (u×B)/c + (αB − β∇×B)/c) and c∇×B = 4πj

j = σeff(E + (u×B)/c + αB/c) ∂B = ∇×(u×B + αB) − η ∇×∇×B with η = η + β ∂t eff eff Two effects:

(1) α − effect: j = σeffαB/c

(2) turbulent diffusivity: β η , ηeff = β = ηT

∂B0 = ∇×(u×B0 + u0 ×B + G) − η∇×∇×B0 ∂t simplifying assumptions: G = 0 , u0 incompressible, isotropic , u = 0 , η = 0 Z t 0 ∂ 0 0 Bk = εklmεmrs (ur Bs)dτ + Bk (t0) t0 | {z } ∂xl δkr δls − δksδlr Z t 0 0 0 0 0 ∂uk 0 ∂Bl ∂ul 0 ∂Bk 0 Ei = u ×B i = ε ijk uj (t) Bl + uk − Bk − ul dτ + Bk (t0) t0 ∂xl ∂xl ∂xl ∂xl Z t 0 0 ∂uk (τ) 0 0 ∂Bk = εijk uj (t) Bl − uj (t)ul (τ) dτ ∂xl ∂xl t0 | {z } | {z } y α y β 1 1 1 isotropic turbulence: α = − u0 ·∇×u0τ∗ = − Hτ∗ and β = u02τ∗ 3 3 3 H helicity , τ∗ correlation time

(http://www.solar-system-school.de/alumni/schrinner.pdf, Schrinner et al. 2007)

∂B Dynamo equation: = ∇×(u×B + αB − η ∇×B) ∂t T • spherical coordinates, axisymmetry • u = (0, 0, Ω(r, ϑ)r sin ϑ) • B = (0, 0, B(r, ϑ, t)) + ∇×(0, 0, A(r, ϑ, t)) α , grad Ω ∂B = r sin ϑ(∇×A)·∇Ω − α∇2A + η ∇2B ∂t 1 T 1 ∂A B p B = αB + η ∇2A with ∇2 = ∇2 − (r sin ϑ)−2 t ∂t T 1 1 rigid rotation has no effect α no dynamo if α = 0  2 2  1 α −dynamo with dynamo number Rα α−term α0  ≈  ∼ 1 α2Ω−dynamo ∇Ω−term |∇Ω|L 2   1 αΩ−dynamo with dynamo number RαRΩ

B out αΒ in poloidal field toroidal field by poloidal field toroidal field by differential rotation; by α-effect differential rotation; ∂Ω/∂r < 0 electric currents electric currents α ∼ cos θ by α-effect by α-effect periodically alternating field, here antisymmetric with respect to equator

Bp Bp

αBp αBt

stationary ﬁeld, here antisymmetric with respect to equator

Consider αΩ–equations locally Cartesian coordinates (x, y, z) corresponding to (θ, φ, r)

α = const, ηT = const, u = (0, Ωz, 0) with Ω = const

Bt = (0, B(x, t), 0), Bp = (0, 0, ∂A(x, t)/∂x) 0 00 00 0 B˙ = ΩA + ηT B , A˙ = αB + ηT A , ˙ = ∂/∂t, = ∂/∂x ansatz (B, A) = (B0, A0) exp[i(ωt + kx)] 2 2 dispersion relation (iω + ηT k ) = ikΩα assume αΩ < 0, e.g. α > 0, Ω < 0 and take k > 0 2 1/2 ω = iηT k −(1+i)|kαΩ/2| (Parker 1955) 2 1/2 1/2 2 growth rate −ωI = −ηT k + |kαΩ/2| ≥ 0 for |kαΩ/2| ≥ ηT k : inductive effects must exceed threshold 1/2 frequency ωR = −|kαΩ/2| < 0: wave propagation in positive x-direction identical result for k < 0 if αΩ > 0 wave propagation in negative x-direction Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Antidynamo theorems Basic electrodynamics Parker’s helical convection Kinematic turbulent dynamos Mean-field theory The solar dynamo Mean-field coefficients Magnetohydrodynamical dynamos and geodynamo simulations Mean-field dynamos Dynamo waves and dynamo number

In general: wave propagates along surfaces of constant rotation (Yoshimura 1975) direction of propagation depends on sign(αΩ) period is geometric mean of (kα)−1 and Ω−1 2 −1 in the critical case period equals (ηT k ) , decreasing with increasing excitation

Dynamo number:

2 2 ˜ R ˜ Ω0R Ω = Ω0Ω, α = α0α,˜ t = ˜t, B = B0B˜, A = RB0A˜, A˜ = A˜ ηT ηT ∂B ∂A = r sin θ(∇×A)·∇Ω + ∆ B and = PαB + ∆ A ∂t 1 ∂t 1

2 α0R Ω0R 1/2 P = RαRΩ = · dynamo number , Bt /Bp ≈ (RΩ/Rα) ηT ηT

α = α0 cos x, ∂uy /∂z = G0 sin x dynamo effects A˙ = P cos xB + A 00, B˙ = sin xA 0 + B00 dynamo equations 2 α0L G0L P = RαRΩ = · dynamo number ηT ηT boundary conditions, L = π/2 x x = 0 : A = B = 0 0 π/2 π x = π : A = B = 0 North Pole Equator South Pole x = π/2 : antisymmetric solution, dipolar : A 0 = B = 0 symmetric solution, quadrupolar : A = B0 = 0 now antisymmetric solution Free decay: A˙ = A 00 and B˙ = B00 ωn t 2 An = e sin nx with ωn = −n , n = 1, 3, 5,... ωn t 2 Bn = e sin nx with ωn = −n , n = 2, 4, 6,...

A˙ = P cos xB + A 00 and B˙ = sin xA 0 + B00 expansion in decay modes (complete, orthogonal, satisfy b.c.)

ωt X ωt X A = e an sin nx and B = e bn sin nx n=1,3,5,... n=2,4,6,... sin x cos nx = 1/2 [sin(n + 1)x − sin(n − 1)x] and cos x sin nx = 1/2 [sin(n + 1)x + sin(n − 1)x] Z π/2 orthogonality relations : sin nx sin mx dx = π/4 δnm 0 2 ωam = P/2(bm−1 + bm+1) − m am, m odd 2 ωbm = 1/2((m − 1)am−1 − (m + 1)am+1) − m bm, m even  a   −1 P/2   a   1     1   b   1/2 −4 −3/2   b   2     2     −    ω  a3  =  P/2 9 P/2   a3   b   3/2 −16 −5/2   b   4     4   .   .   .  . .. . vary P until ωR = 0 : Pcrit

antisymmetric with respect to equator ω ω I I -2097 P<0 28 P>0

493 17

-102 4.5

272 ω R ω -16 -9 -4 -1 -16 -9 -4 -1 R 26 NP 1.33211 9.49303E-02

SP -1.33211 -9.49303E-02 Time Time Pcpublic/schmitt/dynamo/dynewp.f and dynew.f Exercise: ﬁnd critical dynamo numbers for quadrupolar solution, symmetric with respect to equator

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Basic incredients Basic electrodynamics Convection zone dynamos Kinematic turbulent dynamos Overshoot layer dynamos The solar dynamo Interface dynamos Magnetohydrodynamical dynamos and geodynamo simulations Flux transport dynamos Basic solar dynamo incredients

No. 1, 2001 MAGNETIC PARITY SELECTION IN THE SUN 435 Figure 6 presents the results in four frames similar to Figure 4 of the tachocline a-e†ect dynamo operating with a meridional circulation. Figure 6a repeats everything in Figure 4a and also shows the streamlines for meridional circulation. Figure 6b clearly shows how the subsurface equatorward meridional circulation pushes the pattern of the toroidal Ðeld toward the equator compared to that in Figure 4b to produce a double-peak low-latitude butterÑy wing (conÐned primarily below 40¡). The weak poleward branch at high-latitude is the consequence of the a)-dynamo action there in a weakly advective regime. Using a maximum surface Ñow speed of 17 m s~1, we obtain a cycle period of about 10.3 yr. We consider Figure 6b to be a reasonable solar-like butterÑy diagram. Figure 6c shows the poleward drifting of the surface radial Ðeld from mid- to high latitudes together with a weak equatorward drift at low latitudes. This is close to what has been seen in the observed time-latitude diagram for the large-scale di†use Ðeld (Wang et al. 1989; Dikpati & Choudhuri 1995). The observed phase relationship between BÕ or@R/0.7 andBr or@R/1, namely, that the polar Ðeld reverses Differential rotation Helical motion Meridional circulation its sign from positive to negative when the subsurface toroidal Ðeld is negative, is well satisÐed in Figure 6c. This is the most difficult feature for mean Ðeld kinematic dynamos to reproduce, and to our knowledge, no dynamos other than Ñux-transport dynamos can reproduce this phase relationship correctly to such an extent. The equatorward migration of the subsurface toroidal Ðeld and the poleward migration of the surface poloidal Ðeld occur because of a sequence of physical processes as follows: The a) tachocline dynamo is generating both the toroidal and poloidal Ðelds in the tachocline, and both of these components migrate equatorward with the equatorward meridional Ñow there. The toroidal Ðelds annihilate with their opposite-hemisphere counterparts, while the poloidal Ðelds join with their opposite-hemisphere counterparts to form arches bridging the northern and southern hemispheres. The anchoring of these arches in the tachocline is broken by the strong upwelling Ñow at the equator. The poloidal magnetic loops are carried to the surface and drift poleward with the surface Ñow after being separated from their counterparts in the opposite hemispheres. Figure 6d shows, again like Figure 4d, that the bands of right-handed (positive) current helicity are mixed with bands of left-handed helicity but that the left-handed current helicity in this case is more dominant over the right- Dieter Schmitt Hydromagnetic Dynamo Theory handed one compared to the dynamo without meridional circulation. Therefore, meridional circulation helps create a better match of the dynamo-generated current helicity at the overshoot layer with the observed current helicity. The observed maximum surface meridional Ñow speed of 15È20ms~1 (and therefore about 1È2ms~1 Ñow speed at the base from mass conservation) leads to a cycle period of about 11 yr. The scaling law between the cycle period and Ñow speed established in Dikpati & Charbonneau (1999; see eq. [12] there) suggests that a slower (faster) Ñow speed IG F . 6.ÈFlux-transport tachocline dynamo solutions. (a) Prescribed would not only change the cycle period to be larger di†erential rotation pattern (dotted contours), meridional circulation (solid streamlines), and tachocline a-e†ect (shaded region) in our computation (smaller) than 11 yr but also alter the phase relationship domain. (b) Time-latitude diagram for the toroidal Ðeld at the core- between the subsurfaceBÕ and the surfaceBr signiÐcantly. envelope interface. (c) Time-latitude diagram for the surface radial Ðeld. (d) This is true in this class of Ñux-transport dynamo also (plots Current helicity pattern (white is positive, black is negative) at the core- not shown here). Therefore, the Ñux-transport dynamo with envelope interface. a tachocline a-e†ect is competitive with previous dynamos that successfully reproduce the main features of the solar cycle. Introduction Basic incredients Basic electrodynamics Convection zone dynamos Kinematic turbulent dynamos Overshoot layer dynamos The solar dynamo Interface dynamos Magnetohydrodynamical dynamos and geodynamo simulations Flux transport dynamos Convection zone dynamos

10 2 −1 αΩ-dynamo in convection zone, Ω(r) with ∂Ω/∂r < 0, α ∼ cos ϑ, ηT = 10 cm s

(Stix 1976)

Theoretical butterﬂy diagram

(Steenbeck and Krause 1969)

Dieter Schmitt Hydromagnetic Dynamo Theory 446 CHARBONNEAU ET AL. Vol. 527

1995a; Charbonneau et al. 1998b; and references therein). momentum transport is strongly enhanced in the horizontal The transition to the overlying di†erentially rotating direction; otherwise, by the solar age radiative di†usion as envelope occurs across a thin([0.1 R_), approximatelyIntroductionwell as angularBasic momentum incredients transport by the secondary, spherical layer often now called the solar tachoclineBasic electrodynamics, follow- meridionalConvection Ñow broadens zone the dynamos tachocline far beyond its ing the work of Spiegel & Zahn (1992). In what follows we inferred radial width (see also Elliott 1997). They ascribe use this term to refer to the full rotationalKinematic shear turbulent layer near dynamosthis enhancedOvershoot horizontal layer transport dynamos to turbulence driven by the base of the convective envelope, irrespectiveThe of its solar exact dynamoshear instabilityInterface of the dynamos di†erential rotation. Magnetohydrodynamicalposition with respect to the dynamos base of the and core-envelope geodynamo inter- simulationsKitchatinovFlux & transportRu diger (1993,dynamos 1995) have developed an face. The Ðrst reliable determination of the location and analytical model for the (anisotropic) turbulent Reynolds thickness of the tachocline was carried out by Kosovichev stresses arising from the interaction between rotation and ^ (1996), who estimated its thickness to be 0.09 0.04 R_. turbulence in the solar convective envelope, which also DifﬁcultiesAn inference of ofthe SunÏs convection internal rotation proÐle from zoneyields a steady dynamos state di†erential rotation that is character- helioseismic data is illustrated in Figure 1. This shows the ized by a shear layer immediately beneath the convective result of inverting the 2 yr LOWL data using a two- envelope (see Kitchatinov &Ru diger 1995, Fig. 2). As in the dimensional Regularized Least Squares method (see ° 3.1). Spiegel & Zahn model, their tachocline is far thicker than The data were obtained using the LOWL instrument oper- suggested by helioseismology. However, they suggest that ating on Mauna Loa on the island0 of Hawaii. The instru- magnetic stresses, rather than enhanced horizontal viscous Intermittency:ment measures Doppler velocitiesB at h theB SunÏsi surface. coupling, are responsible for keeping the thickness of their From these, the frequencies of global solar p-mode oscil- tachocline in agreement with helioseismic inferences lations (see ° 2) can be deduced. 5 (Ru diger & Kitchatinov 1997; MacGregor & Charbonneau PolarityFrom a dynamical rules: pointB of view,∼ the10 matchingG of a lati- 1999; see also Gough & McIntyre 1998). tudinal gradient in angular velocity in the envelope to a The introduction of a magnetic Ðeld within the tachocline rigidly rotating outer radiative core o†ers a challenging is far from being an ad hoc hypothesis. Based on studies of Magneticproblem in stellar buoyancy hydrodynamics. and Approaching storage this the problem: stability and rise rise of toroidal time magnetic Ñuxcycle ropesÈ length problem from the standpoint of laminar theory, Spiegel & presumed to give rise to sunspot pairs upon emerging Zahn (1992) constructed unmagnetized tachocline models through the photosphereÈit is becoming increasingly clear Rotationunder the assumption law of latitudinal geostrophic balance that the stably stratiÐed layers located immediately beneath and have shown that the helioseismically inferred thickness the convective envelope are the most likely location for the of the solar tachocline can only be reproduced if viscous storage of magnetic Ñux (for a review seeSchu ssler 1996). In Butterﬂy diagram such models, the stability properties of the magnetic Ñux ropes depend sensitively on the dynamical and thermody- namical structure of the storage layer, including its radial extent beneath the envelope. Because of the strong radial shear present in its equatorial region, as revealed by helioseismology, the solar tachocline is also a promising location for the seat of the solar dynamo. Thin-layer mean-Ðeld dynamo models can and have been constructed (see, e.g.,Ru diger & Brandenburg 1995, and references therein). A general property of such models is the dependence of the dynamo period on the thickness of the generating layer, reÑecting the decrease of the magnetic dissipation time with decreasing thickness. A related class of dynamo models are the so-called interface dynamos (Parker 1993; Tobias 1996; Charbonneau & Mac- Gregor 1996, 1997), which are characterized by the spatial segregation of the shear and a-e†ect on either side of the core-envelope interface. Once again the behavior of the resulting dynamo is sensitively dependent on the thickness of the shear layer and relies on the bulk of the layer being located in the upper part of the stably stratiÐed radiative core, rather than within the unstably stratiÐed overlying envelope. It is an intriguing and often unappreciated fact that the FIG. 1.ÈSolar internal rotation proÐle as inferred from 2 yr LOWL latitudinal shear proÐle in the solar convection zone lies data using a 2D RLS inversion. Contours of constant rotation rate are close to the stability limit with respect to global, inviscid shown, the values being in nHz. The trade-o† parameters (see SchouDieter et al. SchmitthorizontalHydromagnetic shear instability. Dynamo Watson Theory (1981) originally \ ~5 \ ~3 1994 for details) werekr 10 , kh 10 ; and the discretized mesh for argued that the solar surface latitudinal di†erential rotation the solution had 101 points in the radial direction and 25 points in the latitudinal direction. Only one quadrant is shown, with the equatorial was in fact at the stability limit. However, Dziembowski & plane along the bottom edge Éand the rotation axis along the left edge. The Kosovichev (1987) considered more realistic rotational pro- inferred rotation proÐle in the other three quadrants can be obtained by Ðles and arrived to the opposite conclusion. More recently reÑection in the equatorial plane and the plane of the axis in this Ðgure. ^ Charbonneau, Dikpati, & Gilman (1999) examined the sta- The narrow midlatitude band of deceleration seen atr/R_ 0.85 is an inversion artifact due to the LOWL duty cycle and mode set being bility of the (weaker) horizontal di†erential rotation proÐle restricted to l ¹ 100; it does not appear in inversions using other data sets within the tachocline and found it to be stable. On the other (see, e.g., Schou et al. 1998). hand, Gilman & Fox (1997) have shown that the simulta- Introduction Basic incredients Basic electrodynamics Convection zone dynamos Kinematic turbulent dynamos Overshoot layer dynamos The solar dynamo Interface dynamos Magnetohydrodynamical dynamos and geodynamo simulations Flux transport dynamos Overshoot layer dynamos

• Favourable dynamo site: storage, reduced turbulent diffusivity, rotation, dynamic α-effect • Dynamo action of magnetostrophic waves (Schmitt 1985): magnetic field layer unstable due to magnetic buoyancy → excitation of magnetostrophic waves in a fast rotating fluid 2 vA /vrot ≈ vmw vA vrot vS mw are helical and induce an electromotive force → electric current parallel to toroidal magnetic field

≡ dynamic α-effect: αhBitor = hu×bitor α not based on convection, applicable to strong ﬁelds +

superposition of most unstable waves: θ -

Aequator Pol

• Dynamo model

(Schmitt 1993) • Disadvantages: overlapping wings, parity, α concentrated near equator

• Flux tube instability: B > Bthreshold (Ferriz-Mas et al. 1994)

Parker (1993), Charbonneau and MacGregor (1997), Zhang et al. (2004): convection zone: η large, α Dynamo on interface between @@R overshoot layer: η small, ∂Ω/∂r, most ﬂux 514 CHARBONNEAU & MACGREGOR Vol. 486

FIG. 9.ÈTypical solution for an interface mode concentrated atDieter low latitudes, Schmitt produced byHydromagnetic using a latitudinal Dynamodependency Theory for the a-e†ect given by eq. (15). The format is identical to Figs.37. and Solution parameters are C\105, n \ 10~2, d \ 0.03, andC \[5. The dynamo mode propagates towards the ) a equator (pole) for negative (positive) values ofC . In dimensional units, this solution has a period 22.3 yr forg \ 1012 cm2 s~1. a E

shear-driven interface mode. Because both types of modes associated butterÑy diagrams for the poloidal and toroidal have markedly di†erent phase relationships, their inter- Ðelds, as on Figures48. and action is largely destructive, leading to the rapid decay of In contrast to the solution shown onFigure 7, the both modes even if the dynamo number is increased indeÐ- dynamo mode is now concentrated at low latitudes, and nitely. This interference is also responsible for the lack of propagates toward the equator (pole) for negative (positive) any hybrid modes with positive growth rates for negative C . The poloidal Ðeld emerges at midlatitudes, and has a values of C . a a 4.3. Interface Modes Concentrated in the Equator The interface mode ofFigure 7 is localized in the polar regions, and is thus associated with the polar radial shear, as opposed to the equatorial shear. It is hardly surprising that this interface mode should be excited preferentially, as (1) the adopted shear proÐle (cf. eqs.[10]È[12]) is such that below the interface the radial shear is stronger at the pole than at the equator, and (2) the a-e†ect is maximal at the pole and vanishes at the equator (cf.° 2.3). Can true interface modes associated with the equatorial radial shear be excited? If one is willing to grant oneself complete freedom to specify the latitudinal dependency of the a-e†ect, then the answer is, not surprisingly, positive. For illustrative purposes, consider the following colatu- dinal dependency for the a-e†ect (cf. eq. [9]): G[sin 4h , n/4 ¹ h3n/4 ; h(h) \ (15) 0, otherwise . Figure 9 shows a typical solution at successive phase intervals of *r \ n/6, in the same format as on Figures37, and for a true interface solution having C \ 105, n \ 10~2, ) d \ 0.03, andC \[5. This solution was obtained using a FIG. 10.ÈButterÑy diagrams for the equatorial interface mode shown eq. (15) for the latitudinal dependency of the a-e†ect, is onFig. 9. The format is identical to Figs. 4 and 8. Comparing this Ðgure to mildly supercritical with p \ 2.23 and frequency u \ 42.3, Fig. 8, notice how the poloidal Ðeld at high latitudes now lags the toroidal and is a true interface dynamo solution.Figure 10 is the Ðeld by a phase interval *r ^ n/2. 518 DIKPATI & CHARBONNEAU Vol. 518

1. a full cycle period of approximately 22 years; The toroidal Ðeld butterÑy diagram (Fig. 7a) does exhibit 2. a time-latitude ““ butterÑy ÏÏ diagram for the toroidal a strong equatorward branch conÐned to low latitudes, as Ðeld exhibiting a strong equatorward branch restricted required by the constraint in equations (2a) and (2b), but a below 30¡ latitude; relatively strong (D103 G) toroidal Ðeld is also present at all 3. a surface poloidal Ðeld that Ðrst appears at midlatitudes. As discussed in ° 3, this is an unavoidable induc- latitude, with a dominant polar migrating branch; tive e†ect associated with the latitudinal shear both within 4. a phase shift of n/2 between the surface polar Ðeld and and below the envelope, and no change in model param- the deep toroidal Ðeld, so that the polar Ðeld changing its eters can do away entirely with this feature. One might of polarity from negative to positive when the toroidal Ðeld is course assume that the low-latitude Ðeld leads to the pro- positive and maximal in intensity; duction of sunspots, while the somewhat weaker Ðeld at 5. a maximum toroidal Ðeld at or immediately below the higher latitudes does not; this, however, represents an interface in the range 104È105 G; and extraneous assumption, even though studies of the stability 6. a high-latitude surface poloidal Ðeld of about 10 G. of toroidal Ñux ropes do indicate that the growth rate of the magnetic buoyancy instability is a rather sensitive function Figure 7 shows interface toroidal and surface poloidal of both the assumed toroidal Ðeld strength and latitude (see, butterÑy diagrams, identical in format to Figure 3, for a e.g., Ferriz-Mas 1996). solar-like solution having a maximum surface Ñow speed of The high-latitude surface poloidal Ðeld is also too strong u 1500 cm s , a turbulent di†usivity in the envelope \ ~1 by an order of magnitude. As discussed in ° 3, the intensiÐ- g0 3 10 cm s , and a source coefficient of magni- T \ ] 11 2 ~1 cation of the surface polar Ðeld is an unavoidable conse- tudes \ 20 cm s~1. This solution has a period T \ 19.8 yr, 0 Introduction Basic incredients quence of Ñux conservation in a geometrically converging maximumBasic toroidal electrodynamics ÐeldConvection at the zonecore-envelope dynamos interface B \ Kinematic turbulent dynamos Overshoot layer dynamos Õ meridional Ñow. The most straightforward way to reduce 10.4 kG, and maximum radial Ðeld at the poleB \ 140 G. The solar dynamo Interface dynamos r this intensiÐcation is to alter the morphology of the circula- MagnetohydrodynamicalIt dynamos satisÐes and geodynamo constraints simulations 1,Flux 3, 4,transport and dynamos 5 listed above. Note that a tion Ñow pattern at high latitudes. While this is certainly dynamo period of exactly 22 years is easily achievable by Flux transport dynamos possible, at this stage in the modeling this again does not 438 DIKPATI & GILMANÐne tuning the circulation Vol. 559 speedu , but given the nature of appear critical, especially in view of the fact that the form of by HaleÏs polarity lawBÕ is nearly always antisymmetric without loss of generality). The poloidal Ðeld above the 0 about the equator. Unfortunately, we do not have any photospherethe is assumed model to be a potential this Ðeld would that satisÐes not be a particularly meaningful exer- means to assess the parity of the solar magnetic Ðeld about the equation ($2[1/r2 sin2 h)A \ 0 up to the source our internal circulatory Ñow is speciÐed in a largely ad hoc the equator before 1600,Durney but certainly the Sun (1995), did not select Choudharisurfacecise at r \ 2.5 (theR, and et beyond more al. that, (1995), the so solar since wind Dikpati the solar and cycle Charbonneau period is not (1999): strictly the ““ even parity ÏÏ (the symmetricBÕ and antisymmetric A) stretches the Ðelds radially. The analytical solution for manner. over the past century since bipolar spots are almost never A or@Rz1 has been given in equation (8) of Dikpati & Choud- seen with the same leading• regeneration polarity at the same time in both ofhuri poloidal (1995),constant who included ﬁeld only at the oddT through terms\ in the22 associ- yr); tilt of the bipolar important active point regions here is that a hemispheres. This particular feature of the solar cycle ated Legendre polynomial series in that expression to yield feature has not received much attention in solar dynamo a symmetricsurfaceA solution about meridional the equator. To allow the Ñow speed comparable with obser- CONCLUSION models. Here we investigate the issue of parity selection in dynamo to choose its favored symmetry, we use the same 6. Ñux-transport dynamo models,close extending to our calculations surfaceexpression (Babcock for A as given by 1961, equation (8) Leighton of Dikpati 1969) to a full spherical shell and comparing results with the & Choudhurivations (1995) above produces the photosphere but a keep dynamo all with a period quite close to that observed parity. the terms in the associated Legendre polynomial series. In this paper we have investigated the property of a Ñux In a full spherical shell• model,rotational the equator is no longer shear a We indemandof tachocline the only the solar smooth matching cycle. of A across the boundary of the computational domain, and to demand the photosphere. transport dynamo model. Following the original suggestion antisymmetricBÕ (and, hence, the odd parity dynamo Initializing both the Ñux-transport dynamos (driven by mode) about the equator• is not physical. The boundary the Babcock-Leighton a-e†ect and by tachocline a-e†ect) conditions at the two polestransport and at the bottom boundary of magneticwith their respective ﬂux odd parity by modes meridional (obtained from the circulation of Babcock (1961), we ascribe the regeneration of the poloi- remain as before. Applying a condition on A at the top respective single hemisphere calculation), we allow the boundary requires some discussiony determines(BÕ is again zero there, dynamos migration to relax for several direction centuries. Figures 9 and and 10 cycle period dal Ðeld to the net dipole moment associated with decaying tilted bipolar regions, the tilt being a consequence of the action of Coriolis forces on primarily toroidal magnetic Ñux ropes buoyantly rising through the solar convective envelope. We mathematically model this process as a source term superÐcially resembling the a-e†ect of mean Ðeld theory. In fact, we attempt to build into our poloidal source term insight gained from rising Ñux rope simulations; in particular, the source term is nonlocal, in that the generation of poloidal Ðeld at the surface is made proportional to the strength of the toroidal Ðeld deeper down at the core- envelope interface. The poloidal Ðeld so generated is trans- ported into the solar interior by a large-scale meridional Dieter Schmitt Hydromagnetic Dynamo Theory circulatory Ñow, which we assume to consist of a single Ñow cell by meridional quadrant spanning the whole convective envelope. To the best of our knowledge, our model also di†ers from FIG. 9.ÈFull spherical shell solution for the Ñux-transport Babcock-Leighton dynamo. (a) Prescribed di†erential rotation pattern (dotted contours), meridional circulation (solid streamlines), and Babcock-Leighton surface a-e†ect location (shaded regions) in our computation domain. (b) Time-latitude all related models recently published in that it makes use of diagram for the toroidal Ðeld at r/R \ 0.7. a solar-like internal di†erential rotation proÐle, i.e., a proÐle characterized by a latitudinal di†erential rotation throughout the convective envelope, matching smoothly across a thin shear layer on a radiative core rotating rigidly at a rate equal to that of the surface midlatitudes. The use of such a solar-like angular velocity proÐle has profound consequences for the operation of this class of Ñux transport FIG. 7.ÈButterÑy diagrams for a solar-like solution. The format and dynamo models. In particular, the global morphology of the contour level spacing is identical to Fig. 3. This solution is produced using solution is largely dominated by the e†ect of the latitudinal parameter valuesu \ 1500 cm s~1,cmss \ 20 ~1, and gT \ 3 ] 1011 shear, while the radial shear near the core-envelope inter- cm2 s~1, corresponding0 toC \ 4.7 ]0104, C \ 4.64, andR \ 348. It is ) S m face contributes primarily to the further intensiÐcation of characterized by a cycle period T \ 19.8 yr, maximum toroidal Ðeld the toroidal Ðeld. The latter is also favored by our inclusion strength below the interface,B \ 10 kG, and maximum surface polar Õmax of a highÈmagnetic di†usivity contrast between the convec- ÐeldBr \ 140 G. Note how the surface polar Ðeld (b) changes sign as the low-latitudemax toroidal Ðeld on (a) is maximal in amplitude. tive envelope and underlying radiative core. Although a Introduction Basic incredients Basic electrodynamics Convection zone dynamos Kinematic turbulent dynamos Overshoot layer dynamos The solar dynamo Interface dynamos Magnetohydrodynamical dynamos and geodynamo simulations Flux transport dynamos Flux transport dynamos

(cf Dikpati)

(Charbonneau 2013)

Dieter Schmitt Hydromagnetic Dynamo Theory No. 1, 2001 MAGNETIC PARITY SELECTION IN THE SUN 435 Figure 6 presents the results in four frames similar to Figure 4 of the tachocline a-e†ect dynamo operating with a meridional circulation. Figure 6a repeats everything in Figure 4a and also shows the streamlines for meridional circulation. Figure 6b clearly shows how the subsurface equatorward

Introduction Basic incredients meridional circulation pushes the pattern of the toroidal Basic electrodynamics Convection zone dynamos Ðeld toward the equator compared to that in Figure 4b to Kinematic turbulent dynamos Overshoot layer dynamos The solar dynamo Interface dynamos produce a double-peak low-latitude butterÑy wing Magnetohydrodynamical dynamos and geodynamo simulations Flux transport dynamos (conÐned primarily below 40¡). The weak poleward branch at high-latitude is the consequence of the a)-dynamo No. 1, 2001Overshoot MAGNETIC layer dynamo PARITY SELECTION with IN THE meridional SUN circulation 435 Figure 6 presents the results in four frames similar to Figure action there in a weakly advective regime. Using a 4 of the tachocline a-e†ect dynamo operating with a merid- maximum surface Ñow speed of 17 m s~1, we obtain a cycle ional circulation. Figure 6a repeats everything in Figure 4a and also shows the streamlines for meridional circulation. period of about 10.3 yr. We consider Figure 6b to be a Figure 6b clearly shows how the subsurface equatorward reasonable solar-like butterÑy diagram. meridional circulation pushes the pattern of the toroidal Ðeld toward the equator compared to that in Figure 4b to Figure 6c shows the poleward drifting of the surface produce a double-peak low-latitude butterÑy wing radial Ðeld from mid- to high latitudes together with a weak (conÐned primarily below 40¡). The weak poleward branch at high-latitude is the consequence of the a)-dynamo equatorward drift at low latitudes. This is close to what has action there in a weakly advective regime. Using a maximum surface Ñow speed of 17 m s~1, we obtain a cycle been seen in the observed time-latitude diagram for the period of about 10.3 yr. We consider Figure 6b to be a large-scale di†use Ðeld (Wang et al. 1989; Dikpati & reasonable solar-like butterÑy diagram.(Dikpati and Gilman 2001) Figure 6c shows the poleward drifting of the surface Choudhuri 1995). The observed phase relationship between radial Ðeld from mid- to high latitudes together with a weak B o andB o , namely, that the polar Ðeld reverses equatorward drift at low latitudes. This is close to what has Õ r@R/0.7 r r@R/1 been seen in the observed time-latitude diagram for the its sign from positive to negative when the subsurface toroi- large-scale di†use Ðeld (Wang et al. 1989; Dikpati & dal Ðeld is negative, is well satisÐed in Figure 6c. This is the Choudhuri 1995). The observed phase relationship between BÕ or@R/0.7 andBr or@R/1, namely, that the polar Ðeld reverses most difficult feature for mean Ðeld kinematic dynamos to its sign from positive to negative when the subsurface toroidal Ðeld is negative, is well satisÐed in Figure 6c. This is the reproduce, and to our knowledge, no dynamos other than most difficult feature for mean Ðeld kinematic dynamos to Ñux-transport dynamos can reproduce this phase relation- reproduce, and to our knowledge, no dynamos other than Ñux-transport dynamos can reproduce this phase relationship correctly to such an extent. ship correctly to such an extent. The equatorward migration of the subsurface toroidal The equatorward migration of the subsurface toroidal Ðeld and the poleward migration of the surface poloidal Ðeld and the poleward migration of the surface poloidal Ðeld occur because of a sequence of physical processes as Ðeld occur because of a sequence of physical processes as follows: The a) tachocline dynamo is generating both the toroidal and poloidal Ðelds in the tachocline, and both of follows: The a) tachocline dynamo is generating both the these components migrate equatorward with the equator- Dieter Schmitt Hydromagnetic Dynamo Theory toroidal and poloidal Ðelds in the tachocline, and both of ward meridional Ñow there. The toroidal Ðelds annihilate with their opposite-hemisphere counterparts, while the po- these components migrate equatorward with the equator- loidal Ðelds join with their opposite-hemisphere counterparts to form arches bridging the northern and southern ward meridional Ñow there. The toroidal Ðelds annihilate hemispheres. The anchoring of these arches in the tacho- with their opposite-hemisphere counterparts, while the po- cline is broken by the strong upwelling Ñow at the equator. The poloidal magnetic loops are carried to the surface and loidal Ðelds join with their opposite-hemisphere counter- drift poleward with the surface Ñow after being separated parts to form arches bridging the northern and southern from their counterparts in the opposite hemispheres. Figure 6d shows, again like Figure 4d, that the bands of hemispheres. The anchoring of these arches in the tacho- right-handed (positive) current helicity are mixed with cline is broken by the strong upwelling Ñow at the equator. bands of left-handed helicity but that the left-handed current helicity in this case is more dominant over the right- The poloidal magnetic loops are carried to the surface and handed one compared to the dynamo without meridional circulation. Therefore, meridional circulation helps create a drift poleward with the surface Ñow after being separated better match of the dynamo-generated current helicity at from their counterparts in the opposite hemispheres. the overshoot layer with the observed current helicity. The observed maximum surface meridional Ñow speed of Figure 6d shows, again like Figure 4d, that the bands of 15È20ms~1 (and therefore about 1È2ms~1 Ñow speed at right-handed (positive) current helicity are mixed with the base from mass conservation) leads to a cycle period of about 11 yr. The scaling law between the cycle period and bands of left-handed helicity but that the left-handed Ñow speed established in Dikpati & Charbonneau (1999; current helicity in this case is more dominant over the right- see eq. [12] there) suggests that a slower (faster) Ñow speed IG F . 6.ÈFlux-transport tachocline dynamo solutions. (a) Prescribed would not only change the cycle period to be larger handed one compared to the dynamo without meridional di†erential rotation pattern (dotted contours), meridional circulation (solid streamlines), and tachocline a-e†ect (shaded region) in our computation (smaller) than 11 yr but also alter the phase relationship circulation. Therefore, meridional circulation helps create a domain. (b) Time-latitude diagram for the toroidal Ðeld at the core- between the subsurfaceBÕ and the surfaceBr signiÐcantly. envelope interface. (c) Time-latitude diagram for the surface radial Ðeld. (d) This is true in this class of Ñux-transport dynamo also (plots better match of the dynamo-generated current helicity at Current helicity pattern (white is positive, black is negative) at the core- not shown here). Therefore, the Ñux-transport dynamo with envelope interface. a tachocline a-e†ect is competitive with previous dynamos the overshoot layer with the observed current helicity. that successfully reproduce the main features of the solar The observed maximum surface meridional Ñow speed of cycle. 15È20ms~1 (and therefore about 1È2ms~1 Ñow speed at the base from mass conservation) leads to a cycle period of about 11 yr. The scaling law between the cycle period and Ñow speed established in Dikpati & Charbonneau (1999; see eq. [12] there) suggests that a slower (faster) Ñow speed IG F . 6.ÈFlux-transport tachocline dynamo solutions. (a) Prescribed would not only change the cycle period to be larger di†erential rotation pattern (dotted contours), meridional circulation (solid streamlines), and tachocline a-e†ect (shaded region) in our computation (smaller) than 11 yr but also alter the phase relationship domain. (b) Time-latitude diagram for the toroidal Ðeld at the core- between the subsurfaceBÕ and the surfaceBr signiÐcantly. envelope interface. (c) Time-latitude diagram for the surface radial Ðeld. (d) This is true in this class of Ñux-transport dynamo also (plots Current helicity pattern (white is positive, black is negative) at the core- not shown here). Therefore, the Ñux-transport dynamo with envelope interface. a tachocline a-e†ect is competitive with previous dynamos that successfully reproduce the main features of the solar cycle. Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals MHD equations of rotating ﬂuids in non-dimensional form

Navier-Stokes equation including Coriolis and Lorentz forces ! ∂u Ra E r 1 E + u·∇u − ∇2u + 2zˆ×u + ∇Π = T + (∇×B)×B ∂t Pr r0 Pm Inertia Viscosity Coriolis Buoyancy Lorentz

Induction equation ∂B 1 = ∇×(u×B) − ∇×∇×B ∂t Pm Induction Diffusion Energy equation ∂T 1 + u·∇T = ∇2T + Q ∂t Pr Incompressibility and divergence-free magnetic ﬁeld

∇·u = 0 , ∇·B = 0

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Non-dimensional parameters

Control parameters (Input) Parameter Deﬁnition Force balance Model value Earth value

Rayleigh number Ra = αg0∆Td/νκ buoyancy/diffusivity 1 − 50Racrit Racrit Ekman number E = ν/Ωd2 viscosity/Coriolis 10−6 − 10−4 10−14 Prandtl number Pr = ν/κ viscosity/thermal diff. 2·10−2 − 103 0.1 − 1 Magnetic Prandtl Pm = ν/η viscosity/magn. diff. 10−1 − 103 10−6 − 10−5

Diagnostic parameters (Output) Parameter Deﬁnition Force balance Model value Earth value Elsasser number Λ = B2/µρηΩ Lorentz/Coriolis 0.1 − 100 0.1 − 10 Reynolds number Re = ud/ν inertia/viscosity < 500 108 − 109 Magnetic Reynolds Rm = ud/η induction/magn. diff. 50 − 103 102 − 103 Rossby number Ro = u/Ωd inertia/Coriolis 3·10−4 − 10−2 10−7 − 10−6

Earth core values: d ≈ 2·105 m, u ≈ 2·10−4 m s−1, ν ≈ 10−6 m2s−1

Non-magnetic hydrodynamics in rapidly rotating system E 1 , Ro 1 : viscosity and inertia small balance between Coriolis force and pressure gradient

−∇p = 2ρΩ×u , ∇× :(Ω·∇)u = 0 ∂u = 0 motion independent along axis of rotation, geostrophic motion ∂z (Proudman 1916, Taylor 1921)

Ekman layer: At ﬁxed boundary u = 0, violation of P.-T. theorem necessary for motion close to boundary allow viscous stresses ν∇2u for gradients of u in z-direction 1/2 Ekman layer of thickness δl ∼ E L ∼ 0.2 m for Earth core

inside tangent cylinder: g k Ω: Coriolis force opposes convection outside tangent cylinder: P.-T. theorem leads to columnar convection cells exp(imϕ − ωt) dependence at onset of convection, 2m columns which drift in ϕ-direction inclined outer boundary violates Proudman-Taylor theorem y columns close to tangent cylinder around inner core inclined boundaries, Ekman pumping and inhomogeneous thermal buoyancy lead to secondary circulation along convection columns:

poleward in columns with ωz < 0, equatorward in columns with ωz > 0 y negative helicity north of the equator and positive one south

H < 0

H > 0

ωz > 0 and < 0 cyclones / anticyclones

Dieter Schmitt Hydromagnetic Dynamo Theory Hydromagnetic ﬂow in planetary cores 199

3.3.3. Taylor’s constraint. The very low geophysical values of E and Ro (see section 2.2) Introduction Equations and parameterssuggest that viscous and inertial effects may be small. If we set E Ro 0 in (2.20), we = = Basic electrodynamics Proudman-Taylor theorem,obtain convection in a rotating sphere and Taylor’s constraint

Kinematic turbulent dynamos A simple geodynamo model 1z V 5 qRaT g ( B) B. (3.36) The solar dynamo Advanced models × =−∇ − + ∇ × × This is called the magnetostrophic approximation. It has a fundamental consequence first Magnetohydrodynamical dynamos and geodynamo simulations Reversals discovered by Taylor (1963). If we take the φ-component of (3.36) ∂5 Vs (( B) B)φ (3.37) Taylor’s constraint =−∂φ + ∇ × × and integrate it over the surface of the cylinder C(s) (see figure 7) of radius s, coaxial with the rotation axis, we obtain × −∇ ∇× × 2ρΩ u = p + ρg + ( B) B/4π magnetostrophic regimeVs dS (( B) B)φ dS. (3.38) = ∇ × × ZC(s) ZC(s) ∇· The cylinder C(s) intersects the outer sphere (r 1) at z zT,zB where zT zB u = 0 , ρ = const ; Ω = ω0ez √1 s2( cos θ). The left-hand side of (3.38) is the= net flow= of fluid out of the cylindrical=− = surface.− For= an incompressible fluid, and with no flow into or out of the ends of the cylinder, Consider ϕ-component and integrate over cylindricalthis surface must be zero, withC the( consequences) that

(( B) B)φ dS 0. (3.39) ∇ × × = ∂p/∂ϕ = 0 after integration over ϕ, g in meridionalZ planeC(s) This is Taylor’s constraint or Taylor’s condition. Z 1 Z 2ρΩ u·dS = ((∇×B)×B)ϕ dS C(s) 4π C(s) | {z } = 0 Z ((∇×B)×B)ϕ dS = 0 (Taylor 1963) C(s)

net torque by Lorentz force on any cylinder k Ω vanishes Figure 7. The Taylor cylinder C(s), illustrated for the cases (a) where the cylinder intersects 2 the inner core, and (b) where s>rib. The cylinder extends from z zT √1 s to z zB, = = − = 2 2 where (a) zB r s and (b) zB zT. From Fearn (1994). = ib − =− B not necessarily small, but positive and negative parts of the integrandq The system has the freedom to satisfy this constraint through a component of the cancelling each other out azimuthal ﬂow that is otherwise undetermined. If we take the curl of (3.36) and use V 0, we obtain ∇· = ∂V violation by viscosity in Ekman boundary layers y torsional oscillationsq Ra( T g) (( B) B). (3.40) − ∂z = ∇ × + ∇ × ∇ × × around Taylor state Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Benchmark dynamo

5 −3 Ra = 10 = 1.8 Racrit , E = 10 , Pr = 1 , Pm = 5 a) b) c) d)

radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.83r0 magnetic field flow

(Christensen et al. 2001)

(Olson et al. 1999) Dieter Schmitt Hydromagnetic Dynamo Theory b

c a

Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Generation of toroidal ﬁeld from poloidal ﬁeld b

(Olson et al. 1999) Dieter Schmitt Hydromagnetic Dynamo Theory c a

(cf Aubert)

8 −5 Ra = 1.2×10 = 42 Racrit , E = 3×10 , Pr = 1 , Pm = 2.5 a) b) c) d)

radial magnetic field radial velocity field axisymmetric axisymmetric at outer radius at r = 0.93r0 magnetic field flow

(Christensen et al. 2001)

Dieter Schmitt Hydromagnetic Dynamo Theory a

Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Comparison of the radial magnetic ﬁeld at the CMB a c

GUFM model Full numerical (Jackson et al. 2000) simulation b d

Spectrally ﬁltered simulation at Reversing dynamo at −5 −4 c E = 3·10 , Ra = 42 Racrit, Pm = 1, Pr = 1 E = 3·10 , Ra = 26 Racrit, Pm = 3, Pr = 1 (Christensen & Wicht 2007) Dieter Schmitt Hydromagnetic Dynamo Theory

d Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Scaling laws

(Christensen and Aubert 2006, Christensen et al. 2009, Christensen 2010)

Dieter Schmitt Hydromagnetic Dynamo Theory January 14, 2008 22:46 GeophysicalJournalInternational gji˙3693

Magnetic structure of numerical dynamos 5

(Aubert et al. 2008) Figure 5. Snapshots from (a): DMFI movie 1 of model C and (b): movie 2 of model T. Left-hand panels: top view. Right-hand panels: side view. The inner (ICB) and outer (CMB) boundaries of the model are colour-coded withDieter the Schmitt radial magneticHydromagnetic field Dynamo (a red Theory patch denotes outwards oriented field). In addition, the outer boundary is made selectively transparent, with a transparency level that is inversely proportional to the local radial magnetic field. Field lines are also colour-coded in order to indicate ez-parallel (red) and antiparallel (blue) direction. The radial magnetic field as seen from the Earth’s surface is represented in the upper-right inserts, in order to keep track of the current orientation and strength of the large-scale magnetic dipole. Colour maps for (a): ICB field from −0.12 (blue) to 0.12 (red), in units of (ρμ)1/2D, CMB field from −0.03 to 0.03, Earth’s surface field from −210−4 to210−4. For (b): ICB field from −0.72 to 0.72, CMB field from −0.36 to 0.36, Earth’s surface field from −1.8 10−3 to 1.8 10−3.

vortices into columns elongated along the ez axis of rotation, due to the Proudman–Taylor constraint. The sparse character of the mag- 3.1.1 Magnetic cyclones netic energy distribution results from the tendency of field lines A strong axial flow cyclone (red isosurface in Fig. 4) winds and to cluster at the edges of flow vortices due to magnetic field ex- stretches field lines to form a magnetic cyclone. Fig. 6 relates DMFI pulsion (Weiss 1966; Galloway & Weiss 1981). Since magnetic visualizations of magnetic cyclones, as displayed in Figs 4 and 5, field lines correlate well with the flow structures in our models, with a schematic view inspired by Olson et al. (1999). A mag- we will subsequently visualize the magnetic field structure alone. netic cyclone can be identified by the anticlockwise motion of field The supporting movies of this paper (see Fig. 1 for time window lines clustered close to the equator, moving jointly with fairly stable and Figs 5–9 for extracts) present DMFI field lines, together with high-latitude CMB flux patches concentrated above and below the radial magnetic flux patches at the inner boundary (which we will centre of the field line cluster. Model C (movie 1, Fig. 5a) exhibits refer to as ICB) and the selectively transparent outer boundary very large-scale magnetic cyclones (times 4.3617, 4.3811), which (CMB). We will first introduce the concept of a magnetic vortex, suggest an axial vorticity distribution biased towards flow cyclones. which is defined as a field line structure resulting from the inter- Inside these vortices, the uneven distribution of buoyancy along ez action with a flow vortex. By providing illustrations of magnetic creates a thermal wind secondary circulation (Olson et al. 1999), cyclones and anticyclones, DMFI provides a dynamic, field-line which is represented in red on Fig. 6. This secondary circulation based visual confirmation of previously published dynamo mech- concentrates CMB flux at high latitudes, giving rise to relatively anisms (Kageyama & Sato 1997; Olson et al. 1999; Sakuraba & long-lived (several vortex turnovers) flux patches similar to those Kono 1999; Ishihara & Kida 2002), and allows the extension of found in geomagnetic field models. Simultaneously, close to the such descriptions to time-dependent, spatially complex dynamo equatorial plane, the secondary circulation concentrates field lines regimes. into bundles and also pushes them towards the outer boundary,where

C 2008 The Authors, GJI Journal compilation C 2008 RAS Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint Kinematic turbulent dynamos A simple geodynamo model The solar dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals Reversals

500 years before midpoint midpoint 500 years after midpoint

(Glatzmaier and Roberts 1995)

Dieter Schmitt Hydromagnetic Dynamo Theory January 14, 2008 22:46 GeophysicalJournalInternational gji˙3693

10 J. Aubert, J. Aurnou and J. Wicht

petition between normal and inverse structures is felt at the CMB, as well as at the surface, through an axial quadrupole magnetic field. As in event E1, the axial dipole is not efficiently maintained by this configuration, leaving mostly equatorial field lines inside the shell, maintained by two equatorial magnetic upwellings (Fig. 12b), with slightly inverse (red) ez orientation. Also similar to event E1, a com- petition between faint normal and inverse magnetic anticyclones can be observed (Fig. 12c) until time 171.5 where inverse structures take precedence and rebuild an axial dipole (Fig. 12d), thus completing the reversal sequence. The DMFI sequence for event E2 highlights the role of magnetic upwellings in a scenario which is broadly consistent with that proposed by Sarson & Jones (1999). In the smaller-scale model C, the influence of magnetic upwellings on the dipole latitude and amplitude is not as clear-cut as in model T. Their appearance are, however, associated with tilting Introduction Equations and parameters Basic electrodynamics Proudman-Taylor theorem, convection in a rotating sphere and Taylor’s constraint of the dipole axis as seen from the Earth’s surface (see upper-right Kinematic turbulent dynamos A simple geodynamo model inserts in movie 1). Thus, we argue that two essential ingredients January 14, 2008 22:46 GeophysicalJournalInternational Theg solarji˙3693 dynamo Advanced models Magnetohydrodynamical dynamos and geodynamo simulations Reversals for the production of excursions and reversals in numerical dynamos are the existence of magnetic upwellings and a multipolar ICB mag- 10 J.Reversals Aubert, J. Aurnou and J. Wicht netic field. This agrees with the models of Wicht & Olson (2004), in which the start of a reversal sequence was found to correlate with petition between normal and inverse structures is felt at the CMB,upwelling as events inside the tangent cylinder. well as at the surface, through an axial quadrupole magnetic field. As in event E1, the axial dipole is not efficiently maintained by this configuration, leaving mostly equatorial field lines inside the shell, maintained by two equatorial magnetic upwellings (Fig. 12b),4 with DISCUSSION slightly inverse (red) ez orientation. Also similar to event E1, a com- petition between faint normal and inverse magnetic anticyclonesUnderstanding can the highly complex processes of magnetic field gen- be observed (Fig. 12c) until time 171.5 where inverse structureseration take in the Earth’s core is greatly facilitated by Alfvén’s the- precedence and rebuild an axial dipole (Fig. 12d), thus completingorem and the frozen-flux approximation, provided one supplies the reversal sequence. The DMFI sequence for event E2 highlightsan imaging method which is adapted to the intrinsically 3-D and the role of magnetic upwellings in a scenario which is broadlytime-dependent con- nature of the problem, and also takes into account sistent with that proposed by Sarson & Jones (1999). diffusive effects. The DMFI technique used in the present study aims In the smaller-scale model C, the influence of magneticat up-achieving this goal, and highlights several magnetic structures: wellings on the dipole latitude and amplitude is not as clear-cutmagnetic as anticyclones are found outside axial flow anticyclones, in model T. Their appearance are, however, associated with tiltingand regenerate the axial dipole through the creation of magnetic of the dipole axis as seen from the Earth’s surface (see upper-rightloops characteristic of an alpha-squared dynamo mechanism. Mag- inserts in movie 1). Thus, we argue that two essential ingredientsnetic cyclones are found outside axial flow cyclones, and concentrate for the production of excursions and reversals in numerical dynamosthe magnetic flux into bundles where significant Ohmic dissipation are the existence of magnetic upwellings and a multipolar ICBtakes mag- place. Our description of magnetic vortices confirms and illus- netic field. This agrees with the models of Wicht & Olson (2004),trates previously published mechanisms, as presented for instance by in which the start of a reversal sequence was found to correlateOlson with et al. (1999). By separating the influence of cyclones and anticyclones, we extend these views to more complex cases where there Figure 12. Theupwelling steps involved events in insideevent E2, the(Aubert a tangentfull reversal cylinder. et of al.the dipole 2008) axis occurring at time 171.32 in movie 2 (model T). is a broken symmetry between cyclonic and anticyclonic motion. Dieter Schmitt Hydromagnetic Dynamo Theory Furthermore, we present the first field line dynamic descriptions of magnetic upwellings, which are created by field line stretching and equatorial field lines of inverse (red) polarity. In this context, faint advection inside flow upwellings. magnetic anticyclones4 DISCUSSION producing poloidal field lines of both po- Our models show that the magnetic structures are robust features larities can beUnderstanding observed, which the highly do not complex have aprocesses net effect of onmagnetic the fieldfound gen- at high (model T) as well as moderately low (model C) values regenerationeration of the axial in the dipole, Earth’s which core in is turn greatly collapses. facilitated The equa- by Alfvén’sof the- the Ekman number. This suggests that they pertain to the Earth’s torial dipoleorem component and the is frozen-flux also bound approximation, to collapse due provided to the in- one suppliescore. Since we only have access to the radial component of the termittent characteran imaging of the method upwellings which iswhich adapted maintain to the it. intrinsically A low 3-Dmagnetic and field at the Earth’s CMB, our description of the magnetic amplitude multipolartime-dependent state, therefore, nature of takes the problem, place in the and whole also takes shell, into accountstructure underlying CMB flux patches in the models is of particular where againdiffusive faint magnetic effects. anticyclones The DMFI technique of both polarities used in the can present be studyinterest. aims Inside the tangent cylinder, short-lived CMB patches of seen at differentat achieving locations this (Fig. goal, 11c). and After highlights time 168.71 several the magnetic normal structures:both polarities can be created by the expulsion of azimuthal flux polarity (blue)magnetic magnetic anticyclones anticyclones are take found precedence, outside axial and regen- flow anticyclones,within a magnetic upwelling. These patches are quickly weakened erate an axialand dipole regenerate in 0.2 the magnetic axial dipole diffusion through times the (Fig. creation 11d). of magneticby the diverging flow on the top of the upwelling, therefore, they do Event E2loops starts characteristicwith two equatorial of an alpha-squared magnetic upwellings, dynamo grow- mechanism.not Mag- last more than a vortex turnover, which is equivalent to 60–300 yr ing from ICBnetic flux cyclones spots of opposite are found polarity, outside ataxial the flow edges cyclones, of adjacent and concentratein the Earth’score (Aubert et al. 2007). The observation of a tangent axial vorticesthe (Fig. magnetic 12a). The flux blue into upwelling bundles where feeds a significant normal polarity Ohmic dissipationcylinder inverse flux patch in the present geomagnetic field (Olson (blue) magnetictakes anticyclone, place. Our description while the red of magnetic upwelling vortices feeds anconfirms in- and& illus- Aurnou 1999; Jackson et al. 2000; Hulot et al. 2002), although it verse polaritytrates (red) previously magnetic anticyclone. published mechanisms, At time 171.327 as presented this com- for instanceis weakly by constrained and not observed with all field regularizations Olson et al. (1999). By separating the influence of cyclones and anticyclones, we extend these views to more complex cases where there C 2008 The Authors, GJI Figure 12. The steps involved in event E2, a full reversal of the dipole axis Journal compilation C 2008 RAS occurring at time 171.32 in movie 2 (model T). is a broken symmetry between cyclonic and anticyclonic motion. Furthermore, we present the first field line dynamic descriptions of magnetic upwellings, which are created by field line stretching and equatorial field lines of inverse (red) polarity. In this context, faint advection inside flow upwellings. magnetic anticyclones producing poloidal field lines of both po- Our models show that the magnetic structures are robust features larities can be observed, which do not have a net effect on the found at high (model T) as well as moderately low (model C) values regeneration of the axial dipole, which in turn collapses. The equa- of the Ekman number. This suggests that they pertain to the Earth’s torial dipole component is also bound to collapse due to the in- core. Since we only have access to the radial component of the termittent character of the upwellings which maintain it. A low magnetic field at the Earth’s CMB, our description of the magnetic amplitude multipolar state, therefore, takes place in the whole shell, structure underlying CMB flux patches in the models is of particular where again faint magnetic anticyclones of both polarities can be interest. Inside the tangent cylinder, short-lived CMB patches of seen at different locations (Fig. 11c). After time 168.71 the normal both polarities can be created by the expulsion of azimuthal flux polarity (blue) magnetic anticyclones take precedence, and regen- within a magnetic upwelling. These patches are quickly weakened erate an axial dipole in 0.2 magnetic diffusion times (Fig. 11d). by the diverging flow on the top of the upwelling, therefore, they do Event E2 starts with two equatorial magnetic upwellings, grow- not last more than a vortex turnover, which is equivalent to 60–300 yr ing from ICB flux spots of opposite polarity, at the edges of adjacent in the Earth’score (Aubert et al. 2007). The observation of a tangent axial vortices (Fig. 12a). The blue upwelling feeds a normal polarity cylinder inverse flux patch in the present geomagnetic field (Olson (blue) magnetic anticyclone, while the red upwelling feeds an in- & Aurnou 1999; Jackson et al. 2000; Hulot et al. 2002), although it verse polarity (red) magnetic anticyclone. At time 171.327 this com- is weakly constrained and not observed with all field regularizations

Dipole moment a

Time (Hoyng et al. 2001, Schmitt et al. 2001)

αΩ dynamo with α ﬂuctuations

expansion of magnetic ﬁeld B into dynamo eigenmodes bi X B(r, t) = ai(t)bi(r) i ∂a X X i = λ a + (1 − a2) N a + F a ∂t i i 0 ik k ik k k k ∂p ∂ 1 ∂2 Fokker-Planck equation: = − Sp + Dp ∂t ∂a 2 ∂a2 p(a) probability distribution of fundamental dipole amplitude a = a0 ∂U drift term: S = Λ(1 − a2)a = − ∂a diffusion term D comprising stochastic effects

Ising-Heisenberg model of N spins Si (ϑ) L = K − P ! = ( ) = ∂ ∂L ∂L εχi Si sin ϑi , cos ϑi , i 1,..., N = − κϑ˙ + √ ∂t ˙ ∂ϑ i X ∂ϑi i τ 1 ˙ 2 K(t) = ϑi 1 X 1 X 2 M(t) = (Ω·S ) = cos ϑ i N i N i X 2 X i i P(t) = γ (Ω·Si ) + λ (Si ·Si+1) i i

(Mori et al. 2013)

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos The solar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Literature

P. H. Roberts, An introduction to magnetohydrodynamics, Longmans, 1967 H. Greenspan, The theory of rotating fluids, Cambridge, 1968 H. K. Moffatt, Magnetic field generation in electrically conducting fluids, Cambridge, 1978 E. N. Parker, Cosmic magnetic fields, Clarendon, 1979 F. Krause, K.-H. Rädler, Mean-field electrodynamics and dynamo theory, Pergamon, 1980 F. Krause, K.-H. Rädler, G. Rüdiger (Eds.), The cosmic dynamo, IAU Symp. 157, Kluwer, 1993 M. R. E. Proctor and A. D. Gilbert (Eds.), Lectures on solar and planetary dynamos, Cambridge, 1994 D. R. Fearn, Hydromagnetic flows in planetary cores, Rep. Prog. Phys., 61, 175, 1998

Dieter Schmitt Hydromagnetic Dynamo Theory Introduction Basic electrodynamics Kinematic turbulent dynamos The solar dynamo Magnetohydrodynamical dynamos and geodynamo simulations Literature continued

M. Ossendrijver, The solar dynamo, Astron. Astrophys. Rev., 11, 287, 2003 P. Charbonneau, Dynamo models of the solar cycle, Living Rev. Solar Phys., 2, 2005, updated 7, 2010 Treatise on geophysics, Vol. 8, Core dynamics, P. Olson (Ed.), Elsevier, 2007

P. H. Roberts, Theory of the geodynamo C. A. Jones, Thermal and compositional convection in the outer core U. R. Christensen and J. Wicht, Numerical dynamo simulations G. A. Glatzmaier and R. S. Coe, Magnetic polarity reversals in the core P. Charbonneau, Solar and stellar dynamos, Saas-Fee Advanced Course, Vol. 39, Springer, 2013

Dieter Schmitt Hydromagnetic Dynamo Theory