Solar and planetary dynamos: comparison and recent developments
UUniversal Heliophysical ProcessesProceedings IAU Symposium No. 257, 2009N. Gopalswamy et al. c (cid:13) Solar and planetary dynamos:comparison and recent developments
K. Petrovay
E¨otv¨os University, Department of AstronomyH-1518 Budapest, Pf. 32., Hungaryemail:
Abstract.
While obviously having a common root, solar and planetary dynamo theory havetaken increasingly divergent routes in the last two or three decades, and there are probably fewexperts now who can claim to be equally versed in both. Characteristically, even in the fine andcomprehensive book “The magnetic Universe” (R¨udiger & Hollerbach 2004), the chapters onplanets and on the Sun were written by different authors. Separate reviews written on the twotopics include Petrovay (2000), Charbonneau (2005), Choudhuri (2008) on the solar dynamoand Glatzmaier (2002), Stevenson (2003) on the planetary dynamo. In the following I will tryto make a systematic comparison between solar and planetary dynamos, presenting analogiesand differences, and highlighting some interesting recent results.
Keywords. magnetic fields, MHD, plasmas, turbulence, Sun: magnetic fields, Sun: interior,Earth, planets and satellites: general
1. Approaches to astrophysical dynamos
Dimensional analysis: mixing-length vs. magnetostrophic balance
Faced with a problem like the dynamo, where the governing equations are well known andthe source of difficulties is their complexity, it is advisable to start by order of magnitudeestimates of the individual terms in the equations. A clear and detailed account of thisis given in Starchenko & Jones (2002).In the case of the Sun, such considerations point to a balance between buoyancy and inertial forces as the determinant of the resulting flow pattern in the convective zone.Indeed, the order of magnitude equality of these terms is one of the basic formulæ of themixing length theory of astrophysical convection, as formulated in the 1950s —so thisbalance is customarily referred to as “mixing length balance” in the dynamo literature.On the other hand, in rapidly rotating rotating systems the Coriolis term dominatesover the inertial term. If the magnetic field is strong enough for Lorentz forces to becomparable to the Coriolis force and the buoyancy, another type of balance known as magnetostrophic balance (or MAC balance) sets in. This kind of balance is now generallythought to prevail in the more “mainstream” planetary dynamos, such as those of Earth,Jupiter, Saturn, and possibly Ganymede.1.2.
Implicit models: Mean field theory
As the smallest and largest structures present in the strongly turbulent astrophysicaldynamos are separated by many orders of magnitude, it is hopeless and perhaps unnec-essary to set the full explicit treatment of all scales of motion as a goal. In this sense, allmodels of astrophysical dynamos are necessarily “mean field models”, not resolving scalessmaller than a certain level and representing the effect of those scales with some effectivediffusivities. Yet the term “mean field theory” is customarily reserved for those models1 a r X i v : . [ a s t r o - ph . S R ] J a n Petrovaywhere even the largest scale turbulent motions, thought to be the main contributors todynamo action, remain unresolved.Mean field theory has remained the preferred theoretical tool of solar dynamo studies.Even the effect of obviously non-mean-field effects can be included in mean field models inthe form of parametrized ad hoc terms. E.g. the emergence of strong magnetic flux loopsfrom the tacholine to the photosphere is now thought by some to be a major contributor tothe α -effect (the so-called Babcock-Leighton mechanism for α ). The motion of magnetizedfluid being highly independent of the rest of the plasma, this may appear to be a casewhere the mean field description is bound to fail —but the problem is circumvented inmean field dynamo models e.g. by the introduction of a non-local α -term (Wang et al.1991) or by using different diffusivity values for poloidal and toroidal fields (Chatterjeeet al. 2004). 1.3. Explicit models: Numerical simulations
In contrast to solar dynamo theory, planetary dynamo studies have traditionally focusedon models where the large scale, rotationally influenced turbulent motions are explicitlyresolved. For a long time such studies were essentially kinematical, prescribing the largescale flows by simple mathematical formulæ that satisfy some more or less well foundedbasic physical expectations, and solving only the induction equation. Such studies canstill significantly contribute to our understanding of dynamos (see e.g. Gubbins 2008 for arecent overview). Yet, from the mid-1990s onwards the rapid increase in computer powerhas made it possible to develop explicit dynamical models (aka numerical simulations)of the geodynamo, and this has become the main trend in planetary dynamo research.
2. Observational constraints
We are separated from the Earth’s outer core by 3000 km of intransparent rock, whilethe top of Sun’s convective zone is directly observable across 1 AU of near-empty space.Although magnetic measurements and seismology in principle allow indirect inferenceson conditions in the outer core, these are both limited to relatively large-scale (sphericalharmonic degree l <
13) magnetic structures, while empirical information on flows in thecore is almost completely lacking.In sharp contrast to this, the brightness of the Sun allows high S/N detection of spectralline profiles, permitting a precise determination of Doppler and Zeeman shifts. This notonly provides a wealth of high-resolution observations of flows and magnetic fields at thetop of the convective zone: the sensitive detection of waves and oscillations in the solarphotosphere also allows a detailed reconstruction of flows and magnetic field patterns inlayers lying below the surface.The different amount of empirical constraints are certainly a major factor in deter-mining the different approaches taken by solar and planetary dynamo studies. The mainshortcoming of mean field models impeding progress is the vast number of possibilitiesavailable for the choice of parameters and their profiles, which renders the formulationof mean field dynamo models for the planets a rather idle enterprise. The good empiricalconstraints in the solar case provide an indispensable support by narrowing down therange of admissible models. It is no coincidence that it was the helioseismic determina-tion of the internal rotation profile around 1990 which led to a resurgence in solar meanfield dynamo theory.At the same time, no MHD numerical simulation of the solar convective zone hasbeen able to recover the observed butterfly diagram, and even reproducing the observeddifferential rotation profile is not trivial (cf. Sect. 6 below). Geodynamo simulations olar and Planetary Dynamos
3. State of matter, stratification
One obvious difference between planetary and stellar dynamos is that while the con-ducting matter in stars is ionised gas, in planetary dynamos it’s conducting liquids. Theconsequences of this are twofold. On the one hand, in planets incompressibility is often agood approximation, whereas in the Sun there many scale heights between the top andbottom of the SCZ and the scale height itself varies by several orders of magnitude. Thisextreme stratification or “stacking” of structures of vastly different scale is one majorobstacle in the way of realistic global simulations of the SCZ.Another consequence of the different state of matter in stellar and planetary dynamosis that molecular transport coefficients or diffusivities (such as viscosity, resistivity or heatdiffusivity) are less accurately and reliably known in planetary interiors, given the morecomplex material structure. Together with the unsatisfactorily constrained thermal state,this means that the exact position, boundaries and even nature of the dynamo shell is indoubt in some planets. In the water giants (Uranus and Neptune) the resistivity of theirelectrolytic mantles is very uncertain, affecting the extent and depth of the dynamo layer.In Mercury, the unknown thermal state of the planet and the unknown amount of lightconstituents make the position and thickness of the liquid outer core rather uncertain.(Evidence for a liquid outer core was recently reported by Margot et al. 2007.)
4. Energetics and importance of chemistry
The energy source of the convective motions giving rise to the dynamo introducesfurther divisions into astrophysical dynamos. Jupiter, Saturn and the Sun are luminousenough to drive vigorous convection in their interiors by thermal effects alone. For terres-trial planets the available remanent heat may be only barely sufficient or even insufficientto maintain core convection. In these objects, chemical or compositional driving may bean important contribution to maintaining the convective state of their cores. (See e.g.Stevenson 2003 for numerical estimates.) In the case of the Earth, the most likely can-didate is the piling up of light constituents like sulphur at the bottom of the outer core,as iron is freezing out onto the inner core and the light elements are locked out of thesolid phase. (Arguments for an analoguous mechanism suggested for Saturn, the “heliumrain” now seem to have been weakened by the new results of Stixrude & Jeanloz 2008.)
5. Boundaries and adjacent conducting flows
One important new contribution of numerical simulations was the realization of thecrucial role that the choice of boundary conditions play in determining the solutions.Indications for this effect had already been found in kinematic dynamo calculations: e.g.the assumed conductivity of the solid inner core in a geodynamo model affects strongly itscapability to maintain a dipole dominated field and the frequency of reversals (Hollerbach& Jones 1993). But the most important effect is due to the thermal boundary conditions.Convection is driven by heat input from below and heat loss from the top of the layer. Interrestrial planets the latter occurs by mantle convection, the efficiency of which is thena fundamental determinant of the behaviour of the dynamo. Indeed, as the timescale ofmantle flows is very long compared to those in the dynamo, in addition to the overall Petrovaymantle convection even the instantaneous convection pattern realized now can have aprofound effect on the flow structure in the geodynamo (Glatzmaier 2002). The mostwidely mentioned possibility why Venus does not seem to support a dynamo is that,lacking plate tectonics, Venus’ mantle convection is less efficient, so its heat flux can betransported conductively throughout the core, and no convective instability arises. Theresulting slower cooling of the planet may have the additional result that no inner corehas solidified yet, depriving Venus even of the alternative energy source for convectionthat an inverse molecular weight gradient could provide (Stevenson 2003).The importance of appropriate boundary conditions is now also recognized in simula-tions of the solar convective zone. Despite repeated attempts with ever more powerfulcomputers, the hydrodynamical simulations of the new millennium had until very recentlynot been able to reproduce the observed solar internal rotation profile, the resulting isoro-tation surfaces being cylindrical rather than conical (e.g. Brun & Toomre 2002). The mostrecent state-of-the-art simulation (Miesch et al. 2008), however, brought a breakthroughin this respect. The breakthrough is not due to the higher computing power available,but to the introduction of a small ( ∼ .
6. Regime of operation
Dimensional and nondimensional parameters
The number of parameters uniquely determining a dynamo is quite limited. The geometryis a spherical shell between radii r in and r out , with thickness d = r out − r in and relativethickness parameter x = d/r out . The shell, rotating with angular velocity Ω, is filled withmaterial of density ρ , characterized by momentum, heat and magnetic diffusivities ν , κ olar and Planetary Dynamos η , respectively. Finally, convection in the shell is driven by the buoyancy flux F fedin at the bottom of the shell (Olson & Christensen 2006). An internal heating (cid:15) may beadded to the list for cases where volumetric heat loss from secular cooling or radioactivedecay is significant.The number of relevant parameters can be reduced further, realizing that the role ofsome of the variables is just to set characteristic scales. The length scale is clearly set by d and the mass cale by ρ . For the timescale it has been traditional for certain theoreticalconsiderations to use the resistive time d /η in dynamo theory. However, it was recentlypointed out by Christensen & Aubert (2006) that for rapid rotators 1 / Ω offers a morerelevant scaling. Nondimensionalizing all parameters by these scales (and ignoring (cid:15) ) weare left with only 4 nondimensional parameters. Following Olson & Christensen (2006)the effective buoyant Rayleigh number can be defined as Ra b = F/ (1 − x ) d Ω . Thenondimensional measures of the diffusivities are the Ekman numbersEk = ν/ Ω d Ek κ = κ/ Ω d Ek m = η/ Ω d (1)Instead of the three Ekman numbers, one Ekman number and two Prandtl numbers (i.e.diffusivity ratios) are more commonly used:Pr = ν/κ Pr m = ν/η (2)The specified parameters then, in principle, determine the solution, i.e. the resultingturbulent flow field and magnetic field. These can be characterized by their respectiveamplitudes v and B , as well as their typical length scale l —say correlation length,integral scale or similar. (For simplicity a single length scale is assumed for both variables,ignoring anisotropy). In fact, these fiducial scales of the solution may even be estimated ondimensional grounds without actually solving the dynamo equations (Starchenko & Jones2002). From l and v , turbulent diffusion coefficients can be estimated using the usual Austausch recipe ∼ lv . For the Sun, Earth and gas giants such estimates, summarized inTable 1, are in rough agreement with both observations and simulation results.As nondimensional measures of v and B we introduce the Rossby and Lorentz numbers:Ro = v/ Ω d Lo = v A / Ω d (3)where v A = B/ ( ρµ ) / is the Alfv´en speed. It arguably makes more sense to use l insteadof d in the above definitions, which leads us to “local” versions of these numbers:Ro l = v/ Ω l Lo l = v A / Ω l (4)Were we to adopt the diffusive time scales d /ν and d /η instead of 1 / Ω, the nondi-mensional measures of v and B would be the kinetic and magnetic Reynolds numbers:Re = dv/ν Re m = dv/η (5)(Again, using a “local” length parameter l instead of d may be more relevant.) Table 1.
Estimated values of the flow velocity, length scale and turbulent diffusivityDynamo v [m/s] l [km] D t ∼ lv [m /s]Sun (deep SCZ): 20 10 Earth: 3 · −
100 20Jupiter 10 − –10 − Let us now consider what the characteristic values of these nondimensional parametersare for astrophysical dynamos and for numerical simulations. Petrovay6.2.
Ekman numbers (i.e. diffusivities)
Ekman numbers are extremely small in astrophysical dynamos, far below the range 10 − –10 − accessible to current numerical simulations. The main obstacle to the further reduc-tion of Ek is that by choosing a realistic Rayleigh number, we fix the input of kinetic andmagnetic energy into the system. In a stationary state energy must be dissipated at thesame rate as it is fed in. The viscous dissipation rate is ∼ ν/λ where λ is the smallestresolved scale —so ν cannot be reduced without also increasing the spatial resolution.The situation is similar for resistive (Ohmic) dissipation.6.3. Reynolds numbers
It is interesting to note that the magnetic Reynolds number Re m is the only parameterinvolving a diffusivity whose actual value can be used in current numerical simulationsfor some planets (including the Earth). Rossby numbers, in contrast, are invariably in-tractably small in planetary dynamos. The situation in the Sun is the reverse: Rossbynumbers are moderate (only slightly below unity in the deep convective zone and actuallyquite high in shallow layers). Reynolds numbers, however, are all exteremely high in thesolar plasma, even in the relatively cool photosphere.The significance of Re m consists in the existence of a critical value Re m , crit below whichno dynamo action is possible. Planetary dynamo simulations have led to the surprisingresult that Re m , crit has a universal value of about 40, independently of the other pa-rameters, specifically of the Prandtl numbers (Christensen & Aubert 2006). This is asurprising result as the expectation in turbulence theory, confirmed in numerical simula-tions of small-scale dynamos, has been that Re m , crit should be a sensitive function of Pr m (Boldyrev & Cattaneo 2004). The solution of this apparent contradiction is not known.Elementary estimates show that the convective velocity resulting even for very slightlysupercritical Rayleigh numbers results in magnetic Reynolds numbers well above Re m , crit .This implies that the conditions for convection and for dynamo action may be virtuallyidentical in astrophysical fluid bodies with a high conductivity (Stevenson 2003). Wehave already seen that the apparent lack of dynamo action in Venus is attributed to thelack of convection in its core. Planets with very poorly conductive fluid layers, however,may obviously be convecting without supporting a dynamo. A case in point may be thewater giants where the dynamo sustaining layer may potentially only extend to a thinsublayer of the convecting water mantle and/or be only slightly supercritical, unable togenerate fields strong enough to reach magnetostrophic equilibrium.6.4. Rossby number (i.e. velocity amplitude)
An important issue is how the Rossby and Lorentz numbers (i.e. nondimensional velocityand magnetic field amplitudes) scale with the input parameters of the dynamo problem.The significance of this is twofold. Firstly, as current numerical simulations cannot di-rectly access the parameter range relevant for astrophysical dynamos, such scaling lawscan be used to extrapolate their results into the physically interesting domain. Secondly,it is of interest to compare the scaling laws derived from simulation results to those pre-dicted by physical considerations based on the concept of magnetostrophic equilibrium.For the Rossby number, the scaling law extracted from simulations vs. the law theo-retically expected for MAG balance are:Ro = 0 .
85 Ra . vs . Ro ∼ Ra / (6)Geometrical factors invoked to explain the slight discrepancy in the values of the expo-nents do not seem to be capable of explaining it (Christensen & Aubert 2006). However, olar and Planetary Dynamos Lorentz number (i.e. magnetic field strength)
The scaling of the Lorentz number extracted from dipole-dominated dynamo simulationswith no internal heating (Olson & Christensen 2006) viz. the MAC scaling following fromthe assumption of unit Els¨asser number (Starchenko & Jones 2002) areLo D ∝ Ra / Lo ∝ Ro l1 / (7)Assuming that the local Rossby number scales similarly to (6), the latter relation mayalso be turned into a scaling with Ra , but its exponent (0 . .
25) is even more discrepantfrom the value yielded by the simulations than in the previous case.
7. Predictive value
In solar dynamo theory there is a widely held belief that predictions for at least thenext activity cycle should be possible. Dozens of methods have been proposed for this.The relatively most successful ones are apparently those based on some measure of so-lar activity or magnetism at the onset of the new cycle.Recently Cameron & Sch¨ussler(2007) convincingly argued that what stands in the background of all such methods isjust the well known Waldmeier effect, relating a cycle’s amplitude to the rate of rise ofactivity towards the maximum. As solar cycles are known to overlap by ∼ is a physical relationship between high-latitudemagnetic fields during the minimum and the amplitude of the next maximum (Choudhuriet al. 2007). Either way, the persistently low activity during the present solar minimumseems to be a strong indication that cycle 24 will be a rather weak one, in contrast towidely publicized claims based on a certain class of solar dynamo models (Dikpati &Gilman 2006).Owing to the much longer timescales of planetary dynamos, the possibility for testabletemporal predictions is limited. Planetary dynamo models can partly make up for thisby the availability of several instances of known planetary dynamos. These inlcude 7active dynamos (Mercury, Earth, Ganymede and the four giant planets) and one extinctdynamo (Mars), with the remaining planets also providing some important constraintsprecisely by apparently not supporting a dynamo. A testbed of a certain type of modellingapproach, then, may be whether it is capable to provide a unified explanatory scheme forall these systems. Recent results suggest that planetary dynamo simulations may indeedprovide such an explanatory scheme.Collecting and homogenizing the results of hundreds of geodynamo simulations, Olson Petrovay Figure 1.
The suggested unified classification scheme of planetary dynamos based on thescalings of Olson & Christensen (2006) & Christensen (2006) find an interesting scaling (or rather, “non-scaling”) behaviour ofthe amplitude of the dipolar component of the resulting magnetic field. As mentionedabove, considering only dipole-dominated dynamos without internal heating the scalingrelationship (7) results for the amplitude of the dipole component. This scaling, however,is not valid for all dynamo models. This is borne out in Figure 1 where the ordinateessentially shows the ratio of the two sides of equation (7). The dipole-dominated casesin this plot will clearly lie along a horizontal line drawn at the ordinate value 0.5, cor-responding to the coefficient in equation (7). This is indeed the case for dynamos wherethe Rossby number is low, i.e. where the driving is relatively weak. However, for Rossbynumbers above a critical value of order 0.1, the amplitude of the dipolar component sud-denly drops and the solution becomes multipolar. In the case of internally heated modelsthe situation is similar, but the transition between the two regimes is more gradual.The critical role of the Rossby number in this respect indicates that the underlyingcause of the eventual collapse of the dynamo field is the increasing importance of in-ertial forces in the equation of motion (measured by the Rossby number). This leadsto a breakup of the relatively regular columnar structures dominating rapidly rotatingconvection —the detailed mechanism of this will be discussed further in the next section.A further interesting finding is that dynamos lying near the top of the Rossby numberrange of dipolar solutions are generally dipole-dominated, but occasionally undergo ex-cursions and reversals. In a turbulent system this type of behaviour is rather plausible,given that parameters like Ro are expected to fluctuate, and such fluctuations may oc-casionally take the system into multipolar regime, causing the dipolar field to collapse,and then reform once fluctuations have taken the system back to the dipolar regime.Olson & Christensen (2006) also attempt to place individual planetary dynamos onthe phase plane of Fig. 1, based on known empirical constraints and theoretical con-siderations. The resulting distribution seems to provide an impressively comprehensiveclassification system for planetary dynamos. Gas giants lie safely in the dipole-dominatedregime, suggesting that their dynamos are in a magnetostrophic state maintaining a non-reversing dipolar field. Earth is found near the limit of the dipole-dominated regime, justwhere a dipolar dynamo known to be subject to occasional reversals is expected to be. Asomewhat surprising conclusion of this scheme is that the planetary dynamo most similarto Earth’s in its behaviour may be Ganymede’s, also expected to maintain a reversingdipole. The fact the dipolar magnetic field amplitude of the water giants is significantly olar and Planetary Dynamos x = 0 .
65, appropriate for the geodynamo.For planets with significantly different shell geometries, especially for those with thinshells (Mercury?; the water giants?), results may prove to be significantly different.Note also that the parameter on the abscissa of Fig. 1 is actually a rather particularkind of Rossby number, involving one particular local turbulence timescale, defined in away similar to the the Taylor microscale. As in current numerical simulations the largestand smallest scales are normally separated by no more than one order of magnitude, thisfine distinction between different turbulent scales is not important. But when it comes toapplication to actual planets, the different turbulent timescales differ by many orders ofmagnitude, so choosing the right scale is critical. Christensen & Aubert (2006) providegood arguments for the choice of the particular form of Ro l used in Fig. 1, but it isstill somewhat disconcerting that combining large-scale turbulent velocities with a muchsmaller length scale, the Rossby number loses its widely used physical interpretationas the ratio of rotational and turbulent turnover timescales. In addition, our limitedknowledge on the spectrum of magnetostrophic turbulence (Zhang & Schubert 2000,Nataf & Gagni`ere 2008) makes it hard to derive a reliable value for the Taylor microscalein planetary dynamos: the Kolmogorov spectrum is just a (not too educated) guess.
8. Characteristic flow and field patterns
It is well known that solar activity phenomena appear collectively, in the form ac-tive regions . These regions are the solar atmospheric manifestations of large azimuthallyoriented magnetic flux loops emerging through the convective zone. Indeed, the devel-opment of extremely successful detailed models of this emergence process, first in thethin fluxtube approximation, then in 3D numerical simulations, was probably the mostspectacular success story of solar dynamo theory in the last few decades.In contrast to our familiarity with the basic magnetic structures determining solaractivity, nothing about analoguous structures in planetary dynamos had been knownuntil very recently. This situation has spectacularly changed with the development ofnew visualization and geometrical analysis techniques, which showed that, just like inthe Sun, the magnetic field in planetary dynamos is highly intermittent (25 % of themagnetic energy residing in just 1.6 % of the volume), and led to the recognition of avariety of field and flow structures (Aubert et al. 2008). The two main classes of suchstructures are magnetic cyclones/anticyclones and magnetic upwellings .Magnetic cyclones and anticyclones are the MHD equivalent of Busse’s columnar struc-tures, known to dominate rapidly rotating hydrodynamic convection. The simulationsindicate that magnetic anticyclones, in particular, play a key role in generating andmaintaining a dipole dominated magnetic field in the upper part of the dynamo layerand above. Near the bottom of the shell the field is invariably multipolar, but the thermalwind-driven upflow in the axis of a magnetic anticyclone amplifies the upward convectedfields by stretching, so that at the top of the shell a predominantly dipolar field results.Magnetic upwellings are buoyancy-driven upflows rising nearly radially through theshell owing to their high velocity. They come in two varieties: polar upwellings, limitedto the interior of the tangent cylinder of the inner core, emerge more or less in parallelwith the cyclonic/anticyclonic structures, while equatorial upwellings cut through those0 Petrovaystructures, disrupting their integrity. As a result, equatorial upwellings (distant cousinesof the emerging magnetic flux loops in the Sun) are capable of interfering with the mainte-nance of an organized dipolar field by the magnetic anticyclones. As the buoyant driving,and in consequence the turbulent velocity (i.e. the Rossby number) is increased, the num-ber and vigour of upwellings increases, until they can occasionally disrupt the dipolarmagnetic field maintained by the magnetic anticyclones. Following such interruptions thedipole may be reformed with a polarity parallel or opposite to its previous polarity: suchevents correspond to magnetic excursions and reversals, respectively. Finally, with thefurther increase of Ro the equatorial magnetic upflows permanently discapacitate theanticyclones, and the dominant dipolar field structure collapses. This is the mechanismin the background of the characteristic Ro -dependence of dynamo configurations shownin Fig. 1 and discussed in the previous section. Thus, dynamo studies now seem to haveelucidated, at least on a qualitative level, the basic mechanisms underlying the mostcharacteristic phenomena of both solar and planetary dynamos.
Acknowledgement
Support by the Hungarian Science Research Fund (OTKA) under grant no. K67746and by the EC SOLAIRE Network (MTRN-CT-2006-035484) is gratefully acknowledged.