H.K. Moffatt- Report on the NATO Advanced Study Institute on magnetohydrodynamic phenomena in...

8/3/2019 H.K. Moffatt- Report on the NATO Advanced Study Institute on magnetohydrodynamic phenomena in rotating fluids

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J. Fluid Mech. (1973) , vol . 57, part 4, p , 625-649Printed i7 t Great Brita ib

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Report on the NATO Advanced Study Institute onmagnetohydrodynamic phenomena in rotating fluidsEdited by H. K. MOFFATT

Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge(Received 27 October 1972)

An Advanced Stu dy Ins ti tute sponsored b y NATO on MagnetohydrodynamicPhenomena in Rota ting Fluids was held in Cambridge, En gland , from 26 J u n eto 4 J u l y 1972. The a im of th e meeting was to provide, throug h inv ited lectures,a system atic account of those aspects of th e geophysics of th e earths core an d ofastrophysics in which b ot h ro tation effects and magnetic effects play an imp ortantpart. I n addition t o th e invited lectures, shorter con tributions and discussionswere s trongly encou raged. The ma in Sessions and th e Session Chairmen were a sfollows:A The earths magnetism an d planetary Sir Ed wa rd Bullard

B Convection, differential rota tion , an d magn eto- D r N. 0.WeissC Stellar an d interstellar magn etism; pulsars Prof. E. SpiegelDE W aves and instabilities influenced by Lorentz D r R . Hide

magnetismhydrodynamics in th e su n

L

1 Boundary layers an d detached shear layers an d Prof. E. R . Bentonspin-up problemsand/o r Coriolis forces

F Dynam o theory Prof. P.H. Roberts These Chairmen, together w ith Professor P .A . Gilman an d D r H. K.Moffatt,who acted as Director of the Institute, constituted the Organising Committee.The sum ma ry of th e m eeting w hich follows ha s been substantially written byDr Weiss (sessions B and C), Dr Hide (A and E), Professor Benton (D) andProfessor Ro be rts (F),whose willing collaboration is gratefully acknow ledged.There were just over a hund red participants, the g reat m ajority of whom wereaccommodated a t Trin ity College, Cambridge, fo r the duration of the m eeting.

I

1. The earths magnetism and planetary magnetismThe earth and Jupiter possess fairly strong global-scale magnetic fields, theevidence in the case of Jupiter being non-thermal radio-emission a t decimetreand dekam etre wavelengths. A weak an d pa tch y magnetic field has recen tly beenfound a t the surface of th e Moon. Space-probe magnetometer measurements havebeen made in th e vicinity of Venus a n d Mars, with essentially negative results.There is no evidence, radio-astronom ical or otherwise, th a t Sa turn , Uranus andNeptune are m agnetic, b u t a s these a re large, rapidly rotating , an d partially or40 F L M 57

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626 H . K . MoJffattwholly fluid planets it is not unlikely that fields will be detected when space-probe meamrements are eventually made.S ir Edward Bullard? opened Session A with a survey of observations of theearths magnetic field, which is subject to irregular temporal variations on timescales ranging from fractions of a second up to tens or even hundreds of millionsof years. The main geomagnetic field, which has its origin in the liquid core ofthe earth, can at the present time be roughly represented by the field of a hypo-thetical centtred and nearly axial dipole with a moment 8 x 1Oz5e.m.u.,givinga surface field of about 0 .5 gauss. The non-dipole part of the main field is a gooddeal weaker a t the earths surface than the dipole field ( - 20 per cent) but itfluctuates more rapidly and the r.m.s. values of the respective contributions ofthese two parts of the field to the geomagnetic secular variation are roughlycomparable. A striking feature of the geomagnetic secular variation is thetendency for the non-dipole field to drift westward at about 0.2 of longitudeper year, which is evidently much faster than any westward drift of the dipolefield. At the core-mantle interface the spherical harmonic coefficients of the fieldfall off rather more rapidly than white noise.

Studies of the magnetization of rocks have extended our knowledge of theearths magnetic field back over geological time. The most remarkable resultof this work is the discovery that the polarity of the dipole field has changed ina complicated and erratic way. Thus while two dozen polarity reversals occurredduring the past 4-5x 106 years, the polarity remained unchanged for as long aninterval as 30 x 1 0 6 years during the Permian (Bullard 1968). With only a fewthousand years as a typical interval of time taken for the dipole moment t ochange sign by first decreasing in magnitude while remaining roughly axial andthen increasing as the new polarity is established, individual reversals are com-paratively sudden events. Two particularly puzzling observational results are( a )an apparent correlation in certain rocks between magnetic polarity and stateof oxidation, and ( b ) he extinction of certain species at the last reversal.

Another property of reversals was dealt with by M . Kono later in the sessionwhen he described some stochastic models of the earths magnetic field duringthe past 107 years. Palaeomagnetic researches have established th at during thisperiod the geomagnetic field had the following properties: ( a ) t was essentiallya dipolar field with i ts axis not much removed from the present rotation axis;( b ) that the observed frequency of the geomagnetic dipole intensities showa Gaussian distribution; and ( c ) hat the lengths of the polarity intervals showa Poisson distribution. These properties can evidently be described in termsof a model involving twenty or so hypothetical dipoles scattered throughout theliquid core of the earth.

Representing the earths magnetism in terms of hypothetical dipoles withinthe core was first attempted more than twenty years ago, as a proposed improve-ment on the traditional representation in terms of spherical harmonics. Thoughuseful for certain purposes, the method can be physically misleading and whencombined with the skin depth arguments appropriate to a solid conductor it ledto the erroneous idea that the geomagnetic secular variation must have its origint Those who contributed lectures a t th e meeting are distinguished by italic type.


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Magnetohydrodynamic phenomena in rotating fluids 627within the top few tens of kilometres of the core. Such arguments do not, ofcourse, apply to a conducting j luid, through which magnetic energy can betransmitted much more efficiently by fluid motions.

Thus, early work on hydromagnetic waves in the earth was concerned withthe upward transmission of magnetic energy in the core and across the core-mantle interface. Several papers on the reflexion and refraction of plane, smallamplitude hydromagnetic waves at a plane rigid surface have now been writtenwith the problem of the geomagnetic secular variation in mind. D. . kilesreviewed these papers, pointing out th at incorrect boundary conditions had beenused in early work, and presented the results of his own recent analysis (Skiles1972).Work is now in progress on geophysically important effects due to rotation,non-uniformity of the basic fields, and the shape of the core-mantle interface.It is now generally accepted that ( a ) he main geomagnetic field is due to

ordinary electric currents within the earth and ( b ) hese currents flow mainly inthe liquid core, where they are generated by a hydromagnetic dynamo processinvolving fluid motions of a fraction of a millimetre per second. Thus, theproblem of the earths magnetism is inextricably linked with that of providinga theory of core motions, and this in turn will be inseparable from developmentsin the hydromagnetics of rotating fluids.R.Hide reviewed this par t of the subject,pointing out that high on the list of geophysical phenomena to be accounted forby an acceptable theory of core motions are (i) he highly variable frequency ofreversals in sign of the geomagnetic dipole and (ii) he comparatively short timescale of irregular variations in the length of the day.

The magnetic flux linkage of a perfect conductor cannot change, so thateffects due to ohmic dissipation of energy are central to dynamo theories, thesubject of SessionF of this conference (see Q 6 ) . However, on the short time scaleof the geomagnetic secular variation, the main body of the core should behavein many respects like a perfect conductor and support free hydromagneticoscillations of various kinds if they can be excited. If the strength of the toroidalmagnetic field in the core is N 100 gauss, then those planetary scale oscillationswhich are characterized by magnetostrophic balance between Coriolis andLorentz torques on individual fluid elements would have periods comparable withthe time scale of the geomagnetic secular variations and they might propagatein a manner reminiscent of the westward drift. These slow oscillations mightalso play a key role in the production of the geomagnetic field by the dynamoprocess. (For reviews and extensive lists of references see Acheson & Hide 1972;Hide & Stewartson 1972; Roberts & Soward 1972 . )

There is another general class of planetary-scale hydromagnetic oscillationsof the core but these have periods that are typically much less than the electro-magnetic cut-off period of a few years associated with the weak conductivity ofthe earths mantle and they are consequently incapable of producing magneticeffects at the earths surface. However, the eddy currents induced in the lowermantle by these fast oscillations might suffice to account for the horizontalstresses at the core-mantle interface that produce irregular changes in the earthsrotation. An alternative suggestion as to the nature of these horizontal stressesis that they are largely due to hydromagnetic interactions between core motions

40-2


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628 H . K . Mofa t tand bumps on the core-mantle boundary with typical horizontal dimensionsup to a few thousand kilometres and vertical dimensions of only a few kilometres.Such bumps cannot ye t be resolved by seismological methods but evidence of theirexistence is provided by the recently discovered correlation between planetaryscale features of the earths gravitational and magnetic fields (Hide &Malin1 9 7 0 ) .

The complexity of the horizontal structure (i.e. temperature, height, etc.) ofthe core-mantle interface could determine the frequency of reversals in sign ofthe geomagnetic dipole and other properties of the field, such as the amplitudesof the non-dipole field in secular variation (for review see Cox & Cain 1972) . Sucheffects will have to be taken into account when formulating realistic models ofcore motions, a procedure which will also require further basic studies of thehydromagnetics of rotating fluids in containers of various shapes. One flowproperty to which particular attention will be paid in these studies is the helicity(the scalar product of the velocity and vorticity field), the importance of whichemerges from the work of Parker and others on kinematic dynamos. A surprisinglysimple relationship between helicity and local heat transfer can be found bycombining the equations of motion of a rapidly rotating fluid with the equation ofthermodynamics (Hide, unpublished).

The agencies that stir the core have not yet been identified with certainty,owing largely to ignorance of the detailed chemical composition and otherproperties of the earths deep interior, especially the radioactivity of the core.In reviewing in some detail the suggestions that mechanical stirring associatedwith the precession of the earths rotation axis and thermal stirring due to radio-active heating or to the release of heat of crystallization might suffice, W . V .R.Malkus outlined some of his own important fluid-dynamical experiments onprecession and recent studies of convection a t the melting point (Malkus 1971) .

The magnetism of the earth took pride of place in Session A, but included inHides lecture was a brief discussion of the magnetic fields of Jupiter and themoon. Variability in the rotation of the Jovian radiosources and of the GreatRed Spot has been interpreted as evidence of a gross torsional oscillation ofJupiters internal layers involving a toroidal magnetic field of over 103 gausswithin the planet. The electric currents responsible for this field and for thecomparatively weak poloidal field of 10 gauss a t the visible surface are probablyproduced by hydromagnetic dynamo action driven by thermal convection inJupiters metallic core, if it is liquid, or even in Jupiters ower atmosphere, if itis deep enough, the convection being maintained by gravitational energy releaseassociated with slow contraction in radius of the planet a t about 0.1. cm per year(for review see Hide 1 9 7 1 a ) .

Permanent magnetization of lunar rocks is evidently responsible for the weak( N 10d3 gauss) and patchy magnetic field found a t the moons surface. Someauthors account for this permanent magnetization by invoking a hydromagneticdynamo in a hypothetic lunar core in order to account for an inducing field ofnot less than 10-2 auss, while others feel that alternative sources of the inducingfield, such as the electric current generated by induction during meteorite impact,have not yet been ruled out and might prove to be important. (For pertinentreferences see various articles in The Moon, 4, 1-268, 1972.)


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Magnetohydrodynamic phenomena in rotating Jluids 6292. Convection, differential rotation, and magnetohydrodynamics in thesun

The suns magnetic field has been described in two recent reviews (Parker1 9 7 0 a ; Weiss 1971) . The main observational features were summarized by N .0.Weiss. The solar cycle is shown most clearly by the behaviour of toroidal fields,emerging in bipolar magnetic regions or sunspots and reversing after 11 years.The distribution of these fields provides the characteristic butterfly diagram.The poloidal field can be observed a thighlatitudes, andreverses, rather erratically,around the time of sunspot maximum. Near the equator, the general field showsa sector structure: activity occurs in preferred latitudes which rotate witha fixed synodic period of 27 days, which can be detected in the interplanetarymagnetic field and in recurrent magnetic storms (Wilcox1971). Detailed measure-ments show a much more complicated structure, with fields varying on scalesdown to 1500 km, while the average field over the whole sun fluctuates from dayto day in much the same way as the fields measured in stars with convectiveenvelopes like Sirius and Vega (Severny 1971).A number of dynamo models have been proposed to explain the solar cycle.All depend on the mechanism first put forward by Parker (1955) , whereby thepoloidal field is drawn out by differential rotation to give a toroidal field fromwhich a poloidal field with the opposite sense is produced by cyclonic motions.Babcock (1961) suggested a phenomenological model which was elaborated byLeighton (1969) , who integrated the dynamo equations and obtained a butterflydiagram.F . Krause and K . H . Radler described a more rigorous treatment (Steenbeck &Krause 1969-see Roberts & Stix 1971; Krause & RSidler 1971) based on meanfield electrodynamics. The averaged vector product of the fluctuations in thevelocity and the magnetic field provides an electromotive force parallel to themean magnetic field (the a-effect), arising from inhomogeneous turbulenceinfluenced by Coriolis forces. By using Bochners theorem it can also be shownthat there is generally an enhanced turbulent resistivity (Krause & Roberts1973) . I n the sun, the radial density gradient forces rising gas (which dominatesthe motion) to expand and so to rotate, clockwise in the northern hemisphere andanti-clockwise in the south. Thus there is a net helicity, or a-effect. Solar dynamosbased on this a-effect plus a radial shear in the angular velocity produce butterflydiagrams in excellent agreement with the observations. These calculations havebeen-elaborated by others (Deinzer & Stix 1971; Roberts & Stix 1972; Stix 1972)who show that migrations of sunspot zones towards the equator requires anangular velocity that decreases with radius. H . Kohler described a solution of thedynamo equations in the solar convective zone, using finite differences to obtaina solution matched to a potential field outside. Botha nd the turbulent diffusivityare calculated from a model of the convective zone. Kohler pointed out thatan oscillatory dynamo could only be obtained if the value of a was several ordersof magnitude less than that predicted by Steenbeck & Krause (1969) .

The a-effect relies on an idealized model of turbulence and it is desirable toreIate dynamo action to convection in the sun. 0.T .Qordon reviewed models in


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630 H . K . Moffuttwhich the magnetic fields were maintained by motion in baroclinically unstableflows. A pole-equator pressure difference can give rise to axisymmetric zonal(heliostrophic) flow which is unstable to baroclinic (Rossby) waves, with pre-dominantly horizontal motions. It has been suggested that these baroclinicwaves are responsible for the sun's equatorial acceleration (Starr & Gilman 1965),though they are stabilized in the presence of a toroidal magnetic field (Gilman1967u,b ) . This motion also possesses helicity and can serve as a dynamo. Gilman(1969u, b ) computed the motion and fields for a model with Cartesian geometry;Gordon (1972)has extended this trea tment to a thin spherical shell. Both modelsuse a truncated spectral representation for horizontal variations while verticalshears are described by introducing two discrete layers. Small seed fields growby dynamo action and then reverse, with a period th at varies from 2 to 1 2 years.Although the treatment is valid only for a stably stratified layer, the modelreproduces the salient features of Babcock's dynamo process.

A more accurate calculation requires a proper understanding of the structureof the sun's convective zone. Recent progress in convection theory was discussedin some detail.D. .Gough described an approach to nonlinear convection theorybased on expansion in horizontal eigenmodes of the linearized problem (Gough,Spiegel & Toomre 1973). For BBnard convection in a Boussinesq fluid singlemode calculations have been carried out up to a Rayleigh number of 1025 . Thebehaviour of the Nusselt number N for convection between rigid boundariesagrees with the predictions of asymptotic theory ( N N (a2Rln (a2R))*,where R sthe Rayleigh number and a he horizontal wavenumber in the cell). When twoor three modes are included, behaviour is more exotic: time-dependent motionappears, with different scales in the boundary layers and the main flow. E . A .Spiegel briefly outlined the results (due mainly to E. Graham and J.Latour)obtained when this technique is applied to a compressible fluid, using theanelastic approximation. The method offers a means of checking the mixing-length theory used in stellar structure. A different approach to nonlinear BBnardconvection was described by D. . Moore: he integrated the equations for two-dimensional motions between free boundaries on a computer and extendedearlier calculations (Veronis 1966) to a Rayleigh number 1000 times the criticalvalue. The horizontal variations are more complex than those allowed in multi-mode calculations and it was found that for convection between free boundariesthe Nusselt number varied as RQ in the viscous regime but that N cc RO'S6 whenadvection dominates diffusionin the vertical plumes.7 At high Rayleigh numberspersistent nonlinear oscillations appeared in flattened cells (Moore & Weiss1972).

R.S. Peckover discussed numerical experiments on the effect of convection onmagnetic fields in a conducting fluid. He assumed a constant temperature field(i.e. zero PQcletnumber) but included the dynamical effect of the magnetic fieldon the motion. Weak fields were swept aside, as expected from kinematic calcula-tions, forming flux ropes between convection cells. When the local magneticenergy density became comparable with the kinetic energy density of the floweuffioiently high Prandtl number.t Gough, Spiegel and Toomre also find N N Rf or free boundaries and high R and


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Magnetohydrodynamic phenomena in rotating Ju id s 631this process was halted and some form of equipartition was achieved. However,the maximum magnetic energy density can be an order of magnitude greaterthan the kinetic energy density of convection. The complicated behaviouroccurring when rotation is also present was shown by J . 0.Murphy . t Whereasa magnetic field or rotation applied separately are known to inhibit convection,it is found that the two together can enhance it. Murphy extended the linearizedtheory (Eltayeb 1 9 7 2 ) )using the mean field approximation, and showed that themagnetic field can relax the constraint imposed by rapid rotation: for fixed valuesof R and of a Taylor number sufficient to inhibit convection there is a range ofvalues of the Chandrasekhar number for which convection ispossible (though theNusselt number is always less than it would be in the absence of both rotationand magnetic fields).3'.H .Bussediscussed the effects of rotation on convection in a spherical system.For a thin spherical shell, rotating slowly, an expansion about the marginalstate shows that the preferred modes are given by sectorial spherical harmonics(PE),orresponding to banana-like cells. The cellular pattern rotates faster thanthe shell itself and higher order calculations give an equatorial acceleration th atincreases with depth. This model has been used to explain differential rotationin the sun (Busse 1970a).For a rapidly rotating thick shell, convection occurs asrolls in a cylinder parallel to the axis of rotation (Roberts 1958; Busse 1970b).This may be relevant to convection in the earth's core and recent experimentsshow good agreement with the theoretical predictions.Convective models for solar differential rotation were reviewed by P. A . Gilmanand related to his recent numerical results for convectively driven differentialrotation (Gilman 1973). These calculations are for a model of convection ina Boussinesq fluid uniformly heated from below in an equatorial annulus ofa rotating spherical shell, and use the mean field approximation. The problem issolved for Rayleigh numbers up to a few times the critical value and Taylornumbers T n the range I02 < T < 106.As T ncreases, the equatorial regionsrotate faster (but with a local equatorial deceleration near the surface). Thisdifferential rotation is maintained primarily through momentum transport inthe cells, rather than by meridional circulation; indeed the latter is polewardrather than towards the equator a t the surface. The cell pattern has a progrademotion and the angular velocity decreases with depth. With a Taylor number of3 x 104 many features of solar convection are reproduced, and the pattern ofmotion appears likely to act as a dynamo. These results were compared with thespherical shell models of Busse (1970a) and Durney (1970, 1971) and also withmodels requiring an equatorward circulation (Kippenhahn 1963; Cocke 1967;Kohler 1970; Durney 1972a, b) .The overall structure of the convective zone appears to be separated intogranules near the surface, supergranules deeper down, and giant cells near itsbase (Bumba 1967 ; Simon & Weiss 1968). I n equatorial regions the giant cellsare elongated parallel to the axis of rotation and have an angular velocity corre-sponding to the 27-day period of the sector structure; but this pattern will notextend to the poles. Magnetic fields are concentrated by the different scales of

Joint work with R. Van der Borght.


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632 H . K . Moffattmotion (Howard & Stenflo 1972) and dynamo action may be caused by a radialshear of angular velocity in the giant cells, coupled with cyclonic motions in thesupergranules (Steenbeck & Krause 1969) .

3. Stellar and interstellar magnetism; pulsarsThe observational and theoretical background to astrophysical magnetic fields

was reviewed by E.A. Spiegel and E.N . Parker, and the later discussions wereconcentrated on pulsars and magnetic stars. Spiegel related observed stellarmagnetic fields to the evolutionary states of stars. Main sequence stars withsurface temperatures less than 8000OK have deep convective envelopes and arerelatively slow rotators. Such stars show calcium emission whose strength corre-lates with the strength of the magnetic fields, if we may generalize from the solarcase. As these cool stars age they lose angular momentum and their magneticfield strengths diminish. These developments are closely related with the ex-tensive hydromagnetic effects of the deep convection zones whose details canbe observed on the sun. Here dynamo action is believed to be all-important.

In the stars with surface temperatures higher than 8000OK, convection nearthe surface is weak. Certain A-type stars with surface temperatures in the range80OO0-l5 000" show spectral peculiarities and are slow rotators. Magnetic fieldshave been measured in about 200 of them, with strengths up to 34 kilogauss(Preston 1 9 7 1 ; Ledoux & Renson 1966). The observed properties of these stars,including field strengths, usually vary periodically with periods of the order ofdays.

In the final stages of evolution, stars contract to small radii. For massesbelow - 1 . 3 solar masses, dense stars can be supported by the pressure ofdegenerate electrons. These are white dwarfs whose radii are N 6000 km. Mag-netic fields have recently been discovered in four of them with strengths N 107gauss and these do not seem related with surface convection (which does occurin some white dwarfs). Crudely speaking, these fields are stronger than in ordinarystars in the ratio of radius squared.

An even more condensed stellar state is the neutron star which is thought tobe formed in the supernova process. The masses of neutron stars cannot be muchlarger than about 1M., but the exact value is uncertain. Their radii are - 10km.The pulsars are thought to be rotating neutron stars and their periods range from30 ms t o about 2 sec. Their periods increase with time and in the case of the CrabPulsar rotational energy is lost a t a rate N 1039erg/s. f the rotational brakingis magnetic, a field strength - 1012gaussis indicated, with comparable fieldstrengths derived for all other pulsars whose spin-down can be detected. As withmagnetic white dwarfs such field strengths scale roughly like radius squared incomparison with ordinary magnetic stars.

Parker reviewed the genera1occurrence of magnetic fields and the developmentof theories of their origin (Parker 1 . 9 7 0 4 .Many astrophysical fields are maintainedby dynamo action as the combined result of non-uniform rotation and cyclonicmotion (i.e. motion with mean helicity, see $ 6 below) which occurs naturally inrotating convecting bodies. The essential physical properties of field generation


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Magnetohydrodynamic phenomena in rotating Jluids 633can be shown by assuming cylindrical polar co-ordinates (w, 5, 2). Then non-uniform rotation v$(w,z ) generates a strong large-scale toroidal fieldB,, accordingto

in the axisymmetric case, where e, is a unit vector in the q5 direction and r , ~ sthe magnetic diffusivity. Similarly, for the toroidal vector potential A,, neglectingmean motion,

where r is a measure of the mean cyclonic velocity component (Parker 1955,1970)which may be identified with the a! of Steenbeck & Krause (1969) . The generalbehaviour of solutions to these equations is easily studied in Cartesian geometryand depends on the product F d ( v + / w ) / d w . he behaviour of the earth's fielddepends critically on the distribution of cyclones (Levy 1 9 7 2 ~ )). The outstandingquestion for the sun is the form of the non-uniform rotation: dynamo wavesmigrating towards the equator require that angular velocity should decreasewith radius.

The galactic magnetic field can be measured by the Zeeman effect and Faradayrotation. It has a magnitude of 3-4 x 10-6 gauss but no unique descriptionemerges from the observations. Once again, a dynamo model can be based ona combination of non-uniform differential rotation and local cyclonic turbulence(Parker 1 9 7 1 ~ )).

L. Mestel discussed the interaction between rotation and magnetic fields inmagnetic stars. The oblique rotator model for magnetic variable stars had beenreferred to by Spiegel, who noted in particular Preston's conclusion (1971) thatthe angle of obliquity x between the magnetic and rotation axes tends to be neareither 0 or &r. Mestel argued that the lack of correlation between spectral type,rotation, and surface magnetic field suggests that the fields of these stars arenot maintained by dynamo action (Krause 1971) but rather are slowly decayingfossil fields (Cowling 1945) . A large-scale stellar field is probably dynamicallystable only if toroidal flux loops link the (observable) poloidal field (Wright 1970,1973) . Dissipation of the energy of internal motions causes the star to rotateabout its axis of maximum moment, yielding high or low obliquity according asthe star is dynamically prolate (toroidal flux dominant) or oblate (poloidal fluxdominant) (Spitzer 1958; Mestel & Takhar 1972). Stars will be visibly magneticas long as flux is not dragged below the surface by rotationally driven circulation(Mestel 1965, 1971b, 1972; Wright 1969); thus the magnetic stars should bemainly slow rotators, as is observed (Preston 1970). Angular momentum mayhave been lost through magnetic coupling with a wind emitted during the earlyconvective phase (Mestel 1968a) , a process we see occurring in the sun with itsdynamo-built surface field. (Such coupling also affects the angle x (Mestel19683;Mestel & Selley 1970; Selley 1973) , though less powerfully than the dissipationprocess,) More probably, loss of angular momentum occurs by direct couplingwith the interstellar gas, a process better described by fluid equations (Chia &Henriksen 1972) than by a single-particle analysis (Havnes & Conti 1971). These


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6 3 4 H . K . Moffattideas also have their counterparts in theories of star formation and of thedynamics of white dwarfs and pulsars (Mestel 1971 a ) .A .Hewish introduced the discussion of pulsars with an account of recentobservations. Successive pulses differ from each other but the mean pulse shapedoes not vary very systematically with period. The beam might be expected tobecome narrower as the period and the radius of the light cone increase and thereis some observational evidence for this. A pulse envelope is frequency dependent,particularly when observed at frequencies below 100 MHz. Individual pulsesare often highly polarized (usually elliptically) but the average linear polarizationshifts continuously and systematically through the envelope. A curious featureis the existence of drifting subpulses, which overtake the main pulse a t driftrates that vary by an order of magnitude, though the gap between successivesubpulses remains roughly constant (Hewish 1970) .

The discovery of neutron stars in 1968 provided a physicists paradise, whichwas described by V .Canuto. With an average density of lOI4g the averageseparation of particles is only lO-I3 em, while the Compton wavelength is onlyone-fortieth of the Larmor radius. The outer solid crust, about 500m deep, isa lattice of nuclei, with a conductivity a million times greater than that of copper.The neutron liquid (probably a super-fluid) surrounds a solid core of neutrons andcharged hyperons, about 2 km in radius (Ruderman 1969 , 1972; Cameron 1970) .The opacity in the surface layers is altered by the strong magnetic field so thatneutron stars cool more quickly than had previously been supposed. It is thoughtthat sudden small changes in the rotation rates of pulsars correspond to star-quakes in either the solid crust or core. J . Bazer discussed the problem of wavepropagation in a magnetoelastic medium. A geometrical theory (analogous t ogeometrical optics) has been developed for the linearized equations (Bazer 1972);nonlinear simple waves have also been investigated (Bazer & Karal 1972) andprogress has been made in studying the development of shocks.J . E. Gunn reviewed electromagnetic pulsar models (Ostriker & Gunn 1969) .The region outside the neutron star is almost a vacuum but surface electricfields can accelerate particles outwards along the lines of force to produce a staticspace charge throughout the magnetosphere. The energy loss resulting fromspin-down can be calculated from the magnetic torque a t the light cylinder andis sufficient to account for the radiated energy. Outside the light cylinder thefield depends on the inclination of the dipole axis. I f the magnetic and rotationalaxes are perpendicular, waves are emitted and these can accelerate particles t oenergies of 1014V. S . C . Michel reported a lack of consensus on radiativemechanisms, though the overall picture was generally accepted. With an aligneddipole, corotating plasma would extend to a shock near the light cylinder, whilea stellar wind streamed out from regions near the pole and a high density plasmawould be formed by shock heating in the equatorial plane.

The Session concluded with a comprehensive review by B. Lehnert of theoreticaland experimental investigations on rotating plasmas (Lehnert 197 l ) ,and a shorttalk by V .G .Endean who suggested that lack of success in attempts to verifyexperimentally Ferraros law of isorotation could be explained only throughrecourse to a fundamental modification of Ohms law in a rotating conducting


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Magnetohydrodynamic phenomena in rotating Jluids 635fluid; in the discussion that followed, few were prepared to accept this conclusion,but it was agreed that there was as yet no convincing experimental verificationof Ferraros law, largely because laboratory conditions do not approximate tothe conditions in which the law may be expected to be valid,

4. Boundary layers, detached shear layers, and spin-up problemsThe hydromagnetic spin-up problem is interesting from the purely magneto-

hydrodynamical viewpoint because it represents such a dramatic example ofhow the Lorentz force can affect the distribution of fluid angular momentumwhen that force is rotational; thus, the entire bulk of an electrically conductingfluid can be made to spin faster as a result of a Lorentz body torque induced byshearing motions that arise in boundary layers. Such flows require, for theirunderstanding, a knowledge of viscous and resistive boundary layers, detachedshear layers, and multiple wave systems (e.g. Alfvkn waves and inertial oscilla-tions). In addition, physical motivation for MHD spin-up is provided by thevarious cosmical bodies with strong magnetic fields that appear to be under-going changes in angular velocity. This session was mainly concerned withsituations of astrophysical interest, although some work related to the earthscore was also reported.E . A . Spiegel began by reviewing the astronomical motivation for studyingspin-down (here the phrase is used in the broad sense to denote any process,whatever the time scale, by which the angular velocity of a rotating fluid bodychanges). Observations of solar type stars (Kraft 1967) suggest that theirrotational velocities decay algebraically with age, perhaps as t-4, sufficientlylate in their evolution (Skumanich 1972) . Since different stars presumably haddifferent initial conditions, it need not follow that any one star follows sucha decay law. However, should this be the case, then nonlinear processes (such asthe dependence of solar wind torque on angular velocity, Durney & Stenflo1972) clearly contribute to the rotational braking, because linear spin-downproceeds exponentially in time (Greenspan & Howard 1963) .Other cosmical bodies thought to be spinning down include the peculiar A starsand pulsars. Since pulsars are thought to be rotating neutron stars with solidinner core and outer crust, enclosing a superconducting liquid, spin-down couldinvolve quantized vorticity, a novel situation.

The electromagnetic torque exerted by the solar wind (Weber & Davis 1967)slows the tenuous convective envelope of the sun, but the nearly steady observedsurface angular velocity implies that a flux of angular momentum is suppliedfrom below; consequently, the radiative core must be spinning down, and thecentral issue in solar spin-down is one of time-scale. Lumped parameter modelsof this process suggest that the half-life of core angular momentum is about thesame as the present age of the sun (which may not be coincidental). Dickescontribution to this question and the oblateness measurement (Dicke 1970)were also reviewed by Spiegel.

A major complicating feature of the sun is the highly stable density stratifica-tion beneath the convection zone. Spin-down in a Boussinesq fluid is, therefore,


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636 H . K . Moffatta relevant idealized problem, and the comparatively large existing literature fornon-magnetic linear flow in a cylinder (Clark1972)was reviewed by A . Clark,withemphasis on the influence of buoyancy on (i) Ekman layers, (ii) sidewall layers,and (iii) interior circulations. On the time-scale for homogeneous Ekman spin-down, only a thin layer of interior fluid spins down (when the stratification isstrong), so a quasi-steady state of non-uniform rotation is realized. A similarresult occurs in a sphere with central gravitational field (Clark, Clark, Thomas &Lee 1971). Should the Prandtl number be small (asit is deep in the sun, becauseof efficient radiative conduction), complete spin-down occurs only on the longtime-scale for Eddington-Sweet circulations (Sakurai, Clark & Clark 1971) .E . R.Benton reviewed the theory of hydromagnetic spin-up for a homogeneouselectrically conducting fluid contained in an insulating circular cylinder withan initially uniform axial applied magnetic field B,. Finite amplitude changes inangular speed were allowed, but the effects of sidewall boundary layers ignored(Benton 1973). With B, first set to zero, the strongly nonlinear hydrodynamiccases of spin-up from a state of rest (Wedemeyer 1964) and spin-down to a stateof rest were discussed and illustrated by a film. Spin-up from rest is distinguishedby a propagating detached shear layer which separates a non-rotating cylindricalcore from fluid forced to rotate by viscous action a t sidewalls and end-plates.It was noted that, in spin-down to rest, sidewall boundary layers are prone t oRayleigh centrifugal instability, and Ekman layers may also be unstable; a re-sultant emission of inertial oscillations into the interior could mix the angularmomentum and hasten the spin-down process (compared to theoretical pre-dictions, which ignore instabilities).

The moderate Rossby number nonlinear theory of Greenspan & Weinbaum(1965) was discussed and a failure of that theory for strongly nonlinear spin-upwas noted. An alternative method (Benton 1973), providing uniform validityin Rossby number, was reported; it predicts that inertial effects will cause bothspin-up and spin-down to lengthen (compared with linear theory). Strongly non-linear spin-down is found to decay toward the asymptote algebraically in timerather than exponentially, which may be suggestive for stellar spin-down.

In hydromagnetic spin-down,Ekmanlayers are replaced by Ekman-Hartmannlayers. Linear theory (Gilman & Benton 1968; Benton & Loper 1969; Loper &Benton 1970) shows how an induced Hartmann current system overcompensatesfor an attendant suppression of Ekman pumping to produce more rapid spin-down than in linear non-magnetic spin-down (Greenspan & Howard 1963).Recent work (Benton& Chow 1972;Benton 1973) on the nonlinear version of thisproblem was summarized. Inertial nonlinearity always retards both spin-up andspin-down, but the magnetic coupling acts in the opposite sense. However, forstrongly nonlinear flow, a weak field is very ineffective in spin-down (becauseitis continuously dispersed laterally by the secondary flow) but highly effectivein spin-up (because of field line stretching by secondary flow). Strong magneticeffects (in the sense [Lorentz force)9 [Coriolis force() dominate all nonlinearprocesses and give rapid spin-up times identical to those fo r spin-down.J.C .R.H u n t described a wide variety of sidewall layers and detached shearlayers that arise in mechanically and electrically driven MHD channel flows at


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Magnetohydrodynamic phenomena in rotating jluids 637large Har tm ann num ber (see H un t & Shercliff 1971, for a review). The stronganalogies (like Taylor-Proudm an con straints, an d control of core flows b yH ar tm an n layers) with non-conducting b u t rotating fluids was made especiallyevident.

A g roup a t Florida S tate Un iversi ty is developing laboratory experiments onrotating magnetohydrodynam ics with liquid metals, where careful measurementsof flow speed an d electric field are planned , using hea ted therm istor beads an dan ultrasonic Doppler velocimeter. W . W .Fowlis described the app aratu s andpointed ou t th a t bo th steady motions driven b y differential rotation of to p a ndbotto m lids an d unsteady spin-up flows are con templated. A long-term aim isth e construction of a laboratory hydrom agnetic dynam o, a project that deservesevery encouragement.D. oper described t h e som ew hat elusive tran sien t m agnetic diffusion region

(Benton & Loper 1969; Loper & Benton 1970; Loper 1973) and then turned tothe effect of boundary conductivity on linear hydromagnetic spin-up (Loper1970a , b ; 1971) . Here a n e xt ra loop of electric current flows into th e bou ndaries(possibly as in th e earth 's m antle) an d furth er couples together th e interior fluidand the boundary. I n this context, th e stren gth of coupling is no t simply theratio of boundary conductivity to that of the fluid but rather the rat io ofboundary conductance to t h a t of th e fluid contained in one Ek m an d epth (this islarge for the e arth !).A laboratory experiment b y A. McEwan t o dem onstrate angular m omentummixing b y turbu lence was described briefly by J . J . Monaghan. Fluid containedin a circular cylinder fitted w ith a stretch ed rub ber mem brane fo r a lid is set inuniform rotation an d th en t h e lid is oscillated vertically a t a preselected frequencyto produce inertial waves which are thought to degenerate resonantly intoturbulence. The subsequent visualization of intense vortices is associated withthe mixing of angular momentum.Since th e linear spin -up of a stratified non-conducting fluid an d a homogeneouselectrically conducting fluid are now quite well understood, a useful next step

is probab ly t o a dd stratification an d hydromagnetic coupling together. H owever,before at tempting that , it is necessary to understand rotating hydromagneticsidewall bou ndary layers (somew hat as in Ingh am 1969, b u t without the restric-tion to small magnetic Reynolds number). Moreover, the possibilities of topo-grap hic features on t h e core-mantle interface (see Session A ) suggests thenecessity for studying sp in-up in unconventional containers. Then th e crucialb u t difficult question of stability of these flows m us t be confronted.

5. Waves and instabilities subject to Coriolis and/or Lorentz forcesHydrom agnetic planetary waves (Hide 1966; Malkus 1967 ; Stewartson 1967)and their counterpart in unbounded systems, namely hydromagnetic inertialwaves (Lehnert 1954) , are characterized b y m agnetostrophic b alance betweenCoriolis and Lorentz torques acting on individual fluid elements. They haveperiods 7 N L2C2/V2 where L is a typical length, C2 the angular speed of basicrotat ion and V the A l f v h speed), which are typically very much longer th an


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638 H. . MoSfattthose of ordinary Alfven waves, Ll V ,and of ordinary planetary or inertial waves,2 2-I. The quantity T is 5 logs for the sun (Lehnert 1954; Gilman 1968),5 109sfor the planet Jupiter, 2 04sfor the Crab Pulsar (Hide 1971a), and ifthe strength of the toroidal magnetic field in the earths liquid core is N 100gaussthen V N 10cms-I and T N 10IOs (300 years), which is comparable with thetime-scale of the geomagnetic secular variation. Hydromagnetic planetary wavesin a thin spherical fluid shell propagate eastward (Hide 1966; Stewartson 1967),but in a thick shell, such as the earths liquid core, both eastward and westwardpropagating waves are possible in principle (Hide 1966; Malkus 1967; Hide &Stewartson 1972) unless the vertical temperature gradient is so highly sub-adiabatic as to inhibit westward-propagating modes (by preventing verticalmotions).

The idea that the westward drift of the non-dipole part of the geomagneticfield relative to the equatorial dipole could be a manifestation of westward-propagating hydromagnetic planetary waves in the core (Hide 1966)gains sup-port from recent theoretical studies, which include investigations of (a) therelationship of hydromagnetic planetary waves to eigenoscillations of a finitebody of fluid, especially thin and thick spherical shells (Malkus1967;Stewartson1967; Stewartson & Rickard 1969; Hide & Stewartson 1972); ( b )effects due tostable density stratification (Hide 1969, 1971b; Kalra 1969; Acheson & Hide1972); (c) propagation and critical layer absorption of hydromagnetic inertialwaves in a non-uniform magnetic field, including so-called valve effects(Acheson 1972a, 1973a; McKenzie 1973; Rudraiah & Venkatachalappa 1972);(d) wave-generation by various types of three-dimensional instability (Acheson19723, 19733; Booker 1972; see also Braginskii 1967; Roberts & Soward 1972);and (e) attenuation mechanisms (Acheson& Hide 1973).

Someof the main resultsof these studies were outlined inR.Hides introductoryremarks, which also included a discussion of the governing equations and basicapproximations. I n the first of the papers that followed K . Stewartson consideredsome of the theoretical difficulties encountered in the analysis of eigenoscilla-tions of thick spherical shells of fluid, pointing out that future progress willrequire basic mathematical studies of hyperbolic partial differential equationsunder boundary conditions of the Dirichlet-Neumann type and emphasizingthe need to establish from first principles the extent to which concepts such asgroup velocity are valid when dealing with oscillations governed by such equa-tions and boundary conditions.J . F . McKenzie andD. .Acheson then dealt withvarious aspects of the propagation of hydromagnetic waves in a rotating fluidwhen the basic magnetic field is non-uniform, including critical layer absorption.Acheson also discussed the stability of an annular system with radial gradientsof basic zonal motion, magnetic field strength and buoyancy, showing thatinstabilities tend to manifest themselves as westwardpropagating hydromagneticinertial waves. I n the final paper J . R. ooker considered the stability of magneto-strophic flows in a sphere. The stability criteria are remarkably similar to thosefound in the annular case. Moreover, non-axisymmetric instabilities, when theyoccur, show the same preference for westward propagation as is found in annularsystems.


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Magnetohydrodynamic phenom ena in rotating ,fluids 6396. Dynamo theory

Dynamo theory was the central theme running through the proceedings. Thestage was set early by Bullard, and by Parker, who reviewed the occurrence andcharacter of cosmical fields. It was stressed that, at least in some instances(certainlyin the case of the earth, and debatably in the case of the sun), the freedecay time, rT= P / 7 , of the large-scale current system is so small that a genera-tion mechanism must be sought. This, of course, provides the motivation fordynamo theory. (Here9 s a typical dimension of the body, and 7 is a magneticdiffusivity which requires further interpretation - ee below. When molecularconduction alone operates, 7 = l / p a , where p is permeability and a s electricalconductivity.)

The mathematical nature of the problem was reviewed by P. H . Roberts. Thesimplest theoretical case is provided by the kinematic dynamo problem, in whichall dynamical questions are ignored, and the electrodynamics alone is studied.The velocityU of the conducting fluid is given. (Here, as usual, we take div U = 0.)The question of regeneration poses a linear (vector) eigenvalue problem for ,8,the growth rate of the field B (see, for example, Roberts 1971b) . Attention isfocused on the growth rate with largest real part, since this determines thelong term behaviour of B . This ,8 may be real (d.c.)or complex (a .c . ) . Conditionsare sought under which B is self-excited, i.e. under which the largest 9 ( p )is positive.

The dynamo will fail if the assumed B is axisymmetric (Cowling 1933), if thefluid conductor V is spherical and the assumed U is nowhere radial (Elsasser1946) , if the rate of strain in the flow is too slight (Backus 1958), or if the fluidvelocity in V is everywhere too small (Childress1969) .A convenient dimension-less measure of the fluid velocity is the magnetic Reynolds number R = %9 / r ,where % is a typical magnitude of U.The first two of the antidynamo theoremsjust stated imply some complexity for the field and/or flow postulated; the lasttwo rule out all ideas of solution by expansion about R = 0. Taken together theyaccount for much of the tortuous development of the subject. For exampledirect numerical integrations of the dynamo equations are necessarily intricate.Moreover, it is found that, purely as a result of too crude a truncation, solutionscan be obtained in situations in which it is known (by the first or second anti-dynamo theorem) that none exists.

I .Lerche described to the meeting one numerical procedure based on thevariational formulation of the dynamo equations (Gibson& Roberts 1967) . Thevariational problem is not self-adjoint, and the true eigenvalue is thereforeneither an upper nor a lower bound for those obtained by approximate trialfunctions.H . Kohler propounded a novel grid-point method with which he plans t o solvethe dynamo equations iteratively (see also $ 2 above). The continuity of Bacross the surface S of V gives rise to a non-local boundary condition, i.e. onewhich connects the field at each point of S to all other points on S. Kohlers treat-ment of this condition was not unrelated to one which R. Thir lby later describedto the meeting, and which led to good theoretical results in test cases.


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640 H . K . MojfattThe difficulty of the non-local boundary condition is entirely avoided by the

method of Bullard & Gellman (1954). The field in V (assumed spherical withcentre 0) is first split into toroidal and poloidal parts, curl T r and curl2Sr re-spectively, where r is the radius vector from 0, and U is written similarly. Thedefining scalar T (and likewise S) is further divided into linear combinations ofthe surface harmonics, P p ( 0 ) inmq5 andPK(0)cos mq5, where ( r ,8,q5)are sphericalpolar co-ordinates. The corresponding coefficients T F and TpC and likewiseS r and SEc)are functions of r alone and are therefore governed by coupledordinary differential equations.

Bullard & Gellman proposed a simple dynamo consisting of a shear Tl and anupwelling SlC but did not obtain convergence; Gibson & Roberts (1969), andLilley (1970), took the calculation further, but without success. Lilley alsoexamined a combination of T,, gs and Sp flows, and obtained apparent conver-gence. P.H.Roberts ( 1972) ook the calculation further, however, and showed thatLilleys dynamo also failed, a conclusion supported a t the meeting by D. ubbins.P. H. Roberts reported apparent convergence for a flow synthesized from the T,,S,, Sp and TF harmonics, and integrated by S. Kumar and himself, using theparallel shooting method. Gubbins, following G. 0.Roberts (reported by P. H.Roberts 1971a), selected an axisymmetric motion consisting of many (n) ells.Solutions proportional t o exp (imq5)exist, for which the dynamo equations involveonly two space dimensions ( r and 0 ) . Gubbins displayed convincingly convergedresults. He concluded that the critical R, based on the maximum flow speed,decreases as n increases, and tends to a non-zero limit. As Bullard observed, thislends added support to the two-scale dynamos described below. Gubbinsalso reported th at the most efficient dynamos were those in which the magneticReynolds number based on the largest poloidal velocity in V was about unity.

The difficulties of direct computation suggested that alternatives should beconsidered. Asymptotic methods have been used with great success and, in fact,provided both of the original existence proofs of the subject (Herzenberg 1958;Backus 1958). The best known is the two-scale method, in which it is supposedthat the flow and magnetic field exist on two widely disparate length-scalesL (


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Magnetohydrodynamic phenom ena in rotating Jluids 641electrodynamics b y F. Krause, K . H. Riidler and M. Steenbeck.? It is easy tosee th a t CP is a linear functio nal of B,bu t t o est imate it a closure app roxim ation isnecessary. Commonly f i s t order sm oothing is adopted,i.e. the fluctuating e.m.f.,

Ux B-U x B,is discarded, a procedure which should be acceptable for small R, or smalluT/L,where T is the correlation time.K . H . Radler , speaking for Krause an d himself, surveyed the achievements andpresent status of mean field electrodynamics. He first asked whether incom-pressible statistically-steady isotropic mirror-symmetric turbulence could re-generate field. H e concluded in the negative, a nd gave an argu me nt by Krause &Robe rts (1973)based on Bochners theorem . (F ur th er ndependent corroborationwas provided in a special case by J. Cilliland & K . Aldridge.) He pointed outfurther that ( in a first approximation) C P = -pcur lB , where ,8 is a constantdetermined in principle by the statistical properties of the turbulence, so t h a tth e diffusivity of th e m ean field is essentially vT = 7 +p. Since p > 0 Krause &Rob erts 1973) it follows t h a t the turbu lent conductivity, gT = 1/,qT, s less th a nth e molecular. Riidler also stressed th e analogy between t h e evolution of m eafifield in a turb ule nt electrically co nducting fluid, an d the behaviour of the m eantemp erature in a turbu lent thermally conducting fluid. I n fa ct, in first-ordersmoothing theory, both are governed by the same dispersion relationship(K rause 1972), an d the ap plication of Bochners theorem b y Krause & Roberts(1973) showed th a t b oth decayed w ith time. Lerche conceded a t th e meetingthat some of his publications (Lerche 1971a, b; Lerche & Low 1971) whichpurported to prove otherwise must be i n error.N ex t Riidler considered w hethe r pseudo-isotropic turbulence (which is isotropicb u t no t m irror-symm etric) could be m ore successful. H e concluded th a t it could,and reminded his audience of the discovery of the a-effect, in which CP = a B(see Steenbeck, K rause & RSidler 1966). Once thi s ter m is included in th e m ean -field induction equa tion, the antidy nam o theorems are seen no t to ap ply to B,an d com paratively simple (e.g. axisymm etric) solutions m ay be found. Althoughthe a-effect is attractive in its simplicity, there is, as Radler emphasized, nopractical method known of realizing pseudo-isotropy, and non-homogeneousflows have to be considered. For example, th e a-effect should arise in a bodyrotating with angular velocity o n the presence of density stratifications ora gradient g in turbulen t intensity. Then, however, other terms proportional to(g B)o n d (0.B)g also arise in b, and these might conceivably act againsta-generation. Riidler reported preliminary resuIts showing that, when includedwith the a-effect, dynamo action becomes easier rather than more difficult.(a tself would be proportional to o g and would therefore be of opposite signin opposite hem ispheres.) Riidler also called att en ti on to th e existence of specialmea n electromotive forces, n o t of a-effect typ e, w hich arise from homogeneousturbulence, axisym m etric with respect t o (say) a rota tion axis. These might alsobe capable of sustaining field.

t Most of the writings by these authors appeared originally in German; they haverecently been translated into English (Roberts & Stix 1971).41 F L M 57


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642 H . K. M o f f a t tMany models of working axisymmetric dynamos incorporating the a-effect

have now been constructed. These fall into three main types: the a2-dynamos,the aw-dynamos, and the aw-dynamos with meridional circulation. First, re-generation of mean field is possible through the action of turbulence alone withoutthe aid of bulk motions. Meridional poloidal (P)ield arises from azimuthaltoroidal ( T ) urrents, which are generated from T fields by the a-effect; similarly,T field arises from P currents created from P ields by the a-effect. Functioningtherefore on a product of two a-processes, the name a2-dynamo s appropriate.Numerical computations indicate th at these dynamos are d.c., and efficient, asjudged by the a-effect Reynolds number R, = E 9 / y T , where &is tpyical magni-tude of a.

I n many astrophysical circumstances, the creation of T field from P fieldthrough the a-effect is swamped by its production by a large-scale toroidalshearing motion, and may be neglected. This is the case when R, is small com-pared with the shear Reynolds number R, = OLP/7,, where 8 s a typicalgradient of shear. The a-effect must still be retained for the creation of P fieldfrom T . The resulting picture is much as Parker ( 1 9 5 5 ) originally presented it,although much more information is now available about the behaviour of models,and this was surveyed a t the meeting. P. H. Roberts proposed as a general, butnot universal, rule that when a d < 0 throughout the northern hemisphere (withrespect to the direction of a),he easiest dynamo to excite is of dipolar parityand is a.c., the patterns of magnetic activity drifting from poles towards theequator, as for Maunders butterfly diagram for the sun. If, on the other hand,a d > 0 in the northern hemisphere, the preferred dynamo is of quadrupole parityand is a.c., the patterns moving polewards from the equator. Some field cyclesand butterfly diagrams were presented both by P. r a u s e and by P. H . Robertswhich supported these statements. The sense of migration is that anticipatedfrom the plane dynamo wave of Parker ( 1 9 5 5 ) .Parker described to the meetinghow the wave propagated; M . St ix presented a generalization which allowed ina crude way for the decrease of a with B expected from dynamical considerations(see below and also Stix 1 9 7 2 ) .

The idea that aw-dynamos might be expected to be a.c. as a rule was dis-puted by Parker, who felt that the &function model of Levy ( 1 9 7 2 a ,b ) could beused to prove otherwise. In support, Stix reported an unpublished applicationof the Gibson-Roberts variational method to that model. He found tha t, ifa d > 0, the d.c. dynamo modes found by Levy were indeed preferred over thea.c.; for aw < 0, however, he discovered that a.c. dynamo action occurred atsmaller Reynolds number, unless the sources of a-effect were placed in lowlatitudes. Parker also cited his layer model of the galactic dynamo (e.g. Parker1 9 7 1 a ) in which a d.c. mode had the lowest dynamo number R,R,, i.e. could bethe most easily excited. Late in the meeting, X . Childress and A .M . Soward dis-cussed yet another layer model in which a.c. dynamo action was favoured.

To add further complications, it can happen (as Radler noted) that the modeof aw-dynamo action easiest to excite is asymmetric, and is then necessarily timedependent, the field drifting round the symmetry axis of a and w. The situationremains unclear, but evidently deserves further attention.


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Mag netohydro dynam ic phenomena in r o ta t ing j l u i d s 643The addition of meridional circulation m has a profound effect on the aw -

dynamos (Braginskii 1 9 6 4 ~ ; .H. Roberts 1972) . If m is sufficiently large, thepreferred dynamo appears to be d.c. If it is in the right sense, the efficiency (asmeasured by the critical dynamo number) may be considerably enhanced.P.H. Roberts reported that it seemed to be generally, but not universally, truethat a d > 0 ( < 0) in the northern hemisphere favoured dynamos of dipolar(quadrupolar)parity. He suggested that meridional circulations might be requiredto account for the general steadiness of the geomagnetic field. D. . o p e rremarked that strong meridional circulations might be expected from the non-linear boundary layer described in his talk.

Asymmetric fields can be excited by the ao-mechanism about as readily assymmetric fields (Krause 1971; Stix 1971; Roberts & Stix 1972) . F . K r a u s e , whosurveyed the models, suggested that they provided a viable explanation ofmagnetic stars. If so, he observed, the generally accepted picture of an obliquerotator would have to be restricted to fields showing reflexion symmetry withrespect to the equatorial plane (see 3 3 ) .

Little was said in the meeting about the other well-known asymptotic approachto the dynamo problem, the small asymmetry limit of Braginskii (1964a , b) .By Cowlings theorem, a completely symmetric B cannot be maintained. If, then,a nearly symmetric field and motion is selected, the marginal R for regenerationshould approach infinity as the degree of asymmetry is reduced to zero. Thislimit, R + CO, allows theoreticians to avail themselves of asymptotic methods;moreover, many cosmical fields show a high degree of axisymmetry, so that theresulting theory is not an obvious museum piece. As developed by Braginskii,the theory led to dynamos of ao-type, with or without meridional circulation.A .M . Soward presented a more general form of the R + CO limit, and showed

how a2-dynamoscould arise, as well as those based on the CO A j effect of Radler(1969 ) . Sowards technique (1972) rests on a mixed Lagrangian-Eulerian frame-work, and is a generalization of the method of Frieman & Rotenberg (1960 ) .It was pointed out by H . K . Mof fa t t that the approach could not be used forvelocity fields with linked streamlines.

The linearity of the kinematic dynamo problem, although providing simplifica-tion, introduces a physical absurdity: if the magnetic Reynolds number exceeds(even only slightly) the critical value for marginal regeneration, the field willgrow indefinitely. In practice, of course, the growth would be halted by theLorentz force, which would modify and reduce the flow (Lenzs law) untilamplification is halted. To describe this self-adjustment mechanism the dynamicalequations must be restored. Further, in order to support the Joule expensesof the field, an energy source must be postulated. This may be chosen to simplifythe theory (as in the models of Moffatt, Thirlby and G. 0.Roberts describedbelow), or it may be given by a physical reason, such as thermal buoyancy (asin the models of Busse, Childress and Soward described below). The resultingtheory defines the magnetohydrodynamic (MHD)dynamo problem.

We have noted that the a-effect is associated with non-mirror-symmetricmotions. A s Parker (1955) recognized, a simple class of such flows consists ofright-handed or (left-handed) screw motions in which the velocity, U, and

41-2


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644 H . K . Moffuttvorticity o= curlu are correlated (or anticorrelated), i.e. U. o 0 (oru . o < 0 ) . On AlfvBns frozen field picture, such a mo tion will create a loopin a line of large-scale field B, an d t he electric current, j = p-l curl B, associatedwith th e corresponding change B in field will be anti-co rrelated (or corre lated),i.e. B. c 0, leading to negative a (or B. > 0, leading t o positive a).Clearly,for given driving force, th e larger B, th e less th e m otions will be able to bend th efield lines, an d the smaller a will be. Th us, althou gh a is independ ent of B for th ekinematic dynam o problem, it is a decreasing fun ctio n of B2 n the MHD dynamo.This provides a regulation mechanism for regeneration. It was incorporatedqualitatively in a plane laye r model which M . Xtix discussed a t th e meeting (seealso Braginskii 1970; Stix 1972) . A different kind of field limiting mechanism,nam ely th e depletion of toroidal flux throu gh sunsp ot eruption , was included inth e m odel of th e solar cycle b y Leigh ton (1969)) which W eiss reviewed.A quantitative model of B-dependent a has been studied by H . K. Moffutt(1970b, 1972) an d was described b y him during th e m eeting. Moffatt first con-sidered general properties of the qu an tit y U.o, he helicity of the flow. I n th eabsence of field and with kinematic viscosity v set zero, it is (when integratedover a fixed container) conserved during t he mo tion; indeed, it is a topologicalinva riant related to th e winding of vortex tu bes a bo ut each other (Moffatt 1968).Moffatt showed t h a t inertial waves in a rotatin g flow possess helicity ; e observedthat ( tak ing $2 upwards) rising waves have negative helicity, and falling havepositive helicity. This suggests that a rand om superposition of (sa y) upw ardlypropagating waves will be associated with dynam o action. The kinem atic theory(Moffatt 1 9 7 0 ~ )hows th a t the a-effect is not isotropic, but 8 is neverthelessproportional t o th e com ponents of B (i.e. gt = A, ,B , ) . The wavelength I of themode with the greatest grow th rat e was located, a nd t he w ay in which A ,, variedas increased through th e dynam o action was studied. It was particularly n otedth at , since th e field was a t all times force-free, th e basic assum ption of the the ory( th a t G = 0 in th e rotating frame) was not violated. If L an d T now refer to th ecorrelation length a nd tim e of th e body force exciting the waves, he found t h a tthe ratio of magnetic to kinetic energy is of order (vQT/v)i(Z/L),t least inconditions which might be expected t o ap ply t o th e earths core.Moffatt emphasized that one of the big uncertainties of the theory is themechanism b y w hich, in th e cosmical context, upw ard prop agating inertial waves(say) are preferentially selected, fo r if there is equal representation of upwardan d downgoing waves th e dynam o will fail. A similar difficulty is encounteredin the other main approach to the m agnetohydrodynamic dynamo , namely thegeneralization of th e nearly-symm etric kinem atic m odel described earlier. He reth e prevailing field an d flow are assumed t o be predom inantly axisymmetric andazim uthal, and th e container is supposed to ro ta te with large ang ular velocityS.Clearly, the first two anti-dynamo theorems apply, and regeneration of thistoroidal field will no t be possible unless th e sym m etry is broken an d/o r poloidalmo tion is provided. Braginskii (1964b,1967u, b )proposed, however, t h a t the basicsta te is dynamically unstable t o asym metric wave-like disturbances, which pro-gress in longitude round the rotation axes. The dynamo feeds its Joule lossesfrom this instability. This idea is not u nrelated to t h e role proposed by Gilman


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Magnetohydrodynamic phenom ena in rotat ing j-luids 645(1969a , b ) and Gordon (1972) for baroclinic instabilities of a zonal flow, whichGordon described to the meeting.

Braginskiis theory was discussed and developed by D. .Acheson and by J . R.Booker a t the meeting, and some of the attendant difficulties were highlighted byK . Stewartson; published work has been recently reviewed by Hide & Stewartson(1972) and by Roberts & Soward (1972) . Some support for the existence of theselarge scale waves in the sun was given by S. T . uess. Hide (1966)has long arguedthat the westward drift of the geomagnetic field is a manifestation of a wavemotion; see $5 above.

Por large values of Q9/,where V = .%/(pp)hs the Alfvh velocity based onthe toroidal field strength .%,he waves divide into two groups; fast inertialwaves and slow MAC waves, where MAC stands for Magnetic-Archimedian(buoyancy)-Coriolis, after the forces which determine the propagation of thewaves. It is the MAC waves which are often thought to provide the necessarya-effect. Nevertheless, it can be demonstrated that, in the state of neutralstability, a vanishes when every MAC wave carrying energy outward is balancedby a similar MAC wave carrying energy inward. In both the Moffatt and theBraginskii approach, the fundamental reason for the asymmetry in wave pro-pagation is lacking. Absorption of energy in the Ekman-Hartmann layer at asolid boundary appears to be too weak. Is critical layer absorption indicated,or will gradients of mean density provide the explanation? Clearly, considerableeffort will be expended in answering this question in the future.The theory of MAC wave propagation can be understood on a theory in whichinertial and viscous forces are discarded, t o give the slow-steady equations.These are subject to a certain consistency condition, first given by Taylor (1963) .When this is satisfied, the equations may be solved for U to within a motionconstant on cylinders coaxial with the rotation vector. Even this may be obtainedby differentiating the consistency condition by time and applying the inductionequation. R. Thir lby described briefly his inconclusive attempts to solve the slowsteady equation and the induction equation simultaneously. Plans for a similarstudy were announced by G.O.Roberts, who showed that the time step forintegration must be short compared with the period of the MAC waves relevantto the grid selected. He noted that this was likely to be a stringent condition inthe numerical work.

3.H . Busse , S.Childress and A .M . Soward described simple layer models ofthermally driven MHD dynamos. The presentations were independent, thoughsimilar in content; in particular, the approaches of Childress & Soward (1972)were very closely related. In each case, the field represented a small (but finiteamplitude) disturbance superimposed on a highly rotating fluid layer close to astate of marginal convection (e.g. Chandrasekhar 1961) . As Childress observed,this permitted a direct application of the two-length scale method, for it is wellknown that the preferred convective cells are on a scale L of order T-kY =9,say, where T is the Taylor number, 4 Q 2 9 4 / v 2 .The critical Rayleigh number R, orconvection is of order T )= O ( E - ~ ) .hus, if the magnetic Reynolds number R isO ( d ) , nd the limit E + 0 (T+ CO) is assumed, the dynamo problem for the layerbecomes simple, in principle. Marginal regeneration is found to occur, as an


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646 H . K . Moffatta.c. dynamo wave, provided the plan-form of the convection is not a roll; themost straightforward case is the rectangle, i.e. two perpendicular rolls of differentamplitude. When R exceeds the critical value, the field grows until a state ofbalance is struck. The case in which the resulting Hartmann number BLY(o/pv)Bis of order unity was given particular attention by Soward in his talk.

The Childress-Soward model is governed by three partial differential equa-tions, first order in t and second order in z , the vertical co-ordinate. These beara striking formal relationship to those governing the self-reversing disk dynamosof Rikitake (1958; see also Cook & Roberts 1970), and it was perhaps here thattheoreticians came closest to taking up the challenge of Bullard in the firstlecture to the meeting, namely that of providing a rational explanation forreversals of the geomagnetic field.

R E F E R E N C E SACHESON,. J. & HIDE, R. 1973 Repts . Prog. Ph ys . 36, 159.ACHESON, . J. 1 97 2a J . F l u id M e c h . 53, 01.ACHESON, . J. 197223 J . F luid Mec h. 52, 29.ACHESON,. J. 1 97 3a J . Fluid M e c h . (to appear).ACHESON,. J. 1973 b (in preparation).BABCOCK,. W. 1961 Astrophya. J . 133, 572.BACKUS,. . 1958 Ann. P h y s . 4, 372.BAZER, . 1 97 2 M o n . N o t . R o y . A s t r . S oc . ( to appear).BAZER, . & KARAL, . 1972 To be published.BENTON, . R. 1 97 3 J . F l u i d M e ch . 57, 337.BENTON, . R. & CHOW,J. H. S. 1972 Physics of F l u i d s (to appear).BENTON, . R. & LOPER,D. E. 1969 J . F lu i d M e c h . 39,561.BOOKER,.R. 1972 In preparation.BRAGINSKII,. I. 1 9 64 a S o vi et y P h y s i cs J E T P 20, 726.BRAGINSKII,. I. 1 9 64 3 S ov ie t P h y s ic s J E T P 20, 462.BRAGINSKII,. I. 1964c Geomag. Aer on . 4, 72.BRAGINSKII,. I. 1967a Geomag. Aer on . 7,323.BRAGINSKII,. I . 1967b Cfeomag. Ae ro n. 7, 51.BRAGINSKII,. I. 1970 Ceomag. Ae ro n. 10, 72.BULLARD,. C. 1968 P h i l . T ra ns . R oy . Soc . A263, 81.BULLARD,. C. & GELLMAN,. 1954 P h i l . T ran s . R o y . Soc . A 247,213.BUMBA,. 1967 Plasm a Phys ics , ed. P. A. Sturrock. Academic.BUSSE, . H. 1970a A s t rophys . J . 159, 29.BUSSE,F.H. 1 97 0b J . F l u i d M e c h. 44, 41.CAMERON, .G. W. 1970 Ann. Rev . As tron. As trophys . 8, 179.CHANDRASEKHAR,. 1961 Hyd rodyn amic and H ydromagnet ic S tabi l ity . Oxford UniversityPress.CHIA,T. T. & HENRIKSEN,. N. 1972 A s t rophys . J . (in Press).CHILDRESS,. 1967 Report A F O S R - 6 7 - 0 1 2 4 , Courant Institute, New York.CHILDRESS,. 1969 Th io ri e magn e'tohydrodynam ique de 1'efJet dyn am o, Report fromCHILDRESS,S. 1 970 J . M a t h . P h y s . 11, 3063.CHILDRESS,. & SOWARD,. M. 1972 P h ys . R ev . Le t t . 29, 837.CLARK, . JR . 9 72 A s t r o p hy s . F l u i d D y n . Report No. 18 University of Rochester, 28 pp.CLARK, . JR , LARK,P. A., THOMAS,. H. & LEE,N. H. 197 1 J . F l u i d M e ch . 45, 31 .COCKE,W. J. 1967 A s t rophys . J . 150, 1041.COOK,.E. & ROBERTS,. H. 1970 Proc. Camb. Ph i l . Soc. 68, 47.COWLINU,. G. 1 93 3 M o n . N o t . R. A s t r . Soc . 94, 9.COWLING,. G. 1945 M o n . Not. R. A s t r . Soc . 105,166.

DBpartment MBcanique de la FacultB des Sciences, Paris.


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Mugnetohydrociynamic phenomenu in rotating fluids 647COX,A. & CAIN, . 1972 T r a n s . Am. Geophys . Un. 53, 591.DEINZER,W. & STIX,M. 1971 Astron . As trophys . 12, 111.DICKE,R. H. 1970 Ann. Rev. As tron . & Astrophys . 8, 297.DURNEY, . 1970 Astrophys . J . 161, 1115.DURNEY, . 1971 Astrophys . J . 163, 353.DURNEY, . R. 1972a In Proceedings of the Asilomar Solar Wind Conference (ed. C. P.DURNEY, . 1972b So lar Phy s . 26, 3.DURNEY, . & STENFLO,. 0. 1972 Astrophys . & Space Sc i . 15, 307.ELSASSER, . M. 1946 P h y s . R e v. 69, 106.ELTAYEB,.A. 1972 Proc. Ro y. Soc. A326, 229.FRIEMAN, . & ROTENBERG,. 1960 R e v . M o d . P h y s . 32, 898.GIBSON, . 5 . & ROBERTS,. H. 1967 In Magnetism and the Cosmos. (ed. W. R . Hind-GILMAN, .A. 1 9 6 7 a J . A t m o s . S c i . 24, 101.GILMAN, . A. 1967b J . Atrnos. Sci. 24, 333.GILMAN,. A. 1968 Science 160, 760.GILMAN, . A. 1 9 6 9 a So lar Phys . 8, 316.GILMAN,. A . 19692, Solar Phys. 9, 3.GILMAN,. A. 1973 Solar P h y s . (to appear).GILMAN, .A. & BENTON, . R. 1968 P h y s . F l u i d s. 11, 2397.GORDON, . T. 1972 Design of a numerical solar dynamo model, M.I.T. Departmen1 ofGOUGH, . O ., SPIEGEL, . A. & TOOMRE,. 1973 To be published.GREENSPAN,. P. & HOWARD,. N. 1963 J . Flu id M ech. 17, 385.GREENSPAN,. P. & WEINBAUM,. 1965 J . M a t h . P h y s . 44, 66.HAVNES, . & CONTI, P. S. 1971 Astron . and As trophys . 14, 1.HERZENBERG,. 1958 Phil. T r a n s . R o y . S o c . A250, 543.HEWISH,A. 1970 Ann. Rev . As t r on . As t r ophys . 8 , 265.HIDE,R. 1966 P h i l . T r a n s . R o y . S o c . A 259, 615.HIDE,R. 1969 J . Flu id Mech . 39, 283.HIDE, R. 1 9 7 1 a N a t u r e , 229, 114.HIDE, R. 1971 b Lectures in A p p l i e d M a t h . 13, 229.HIDE,R. & MALIN,S. . C. 1970 N a t u r e , 225, 605.HIDE,R. & STEWARTSON,. 1972 Rev. Geophys . Space P hys . 10, 579.HOWARD, . & STENFLO,. 0. 1972 Solar Phys. 22, 402.HUNT,. C. R. & SHERCLIFF,.A. 1971 Ann. Rev . F lu id Mech. 3, 37.INGHAM,. B. 1969 P h y s . F l u i d s, 12, 389.KALRA,. L. 1969 P u b l. A s t r m . So c. J a p a n , 21, 263.KIPPENHAHN,. 1963 Astrophys . J . 137, 664.KOHLER,H. 1970 So lar Phy s . 13, 3.KRAFT,R. P. 1967 Astrophys . J . 150, 551.KRAUSE, . 1971 As tr on . Nach . 293, 187.KRAUSE, . 1972 As tr on . Nach . 294, 8-3.KRAUSE, . & RADLER,K.H. 1971 Solar Magnetic Fields (ed. R.Howard), p. 770.Reidel, Dordrecht.KRAUSE, . & ROBERTS,. H. 1973 Astrophys . J . (to appear).LEDOUX, . & RENSON,. 1966 Ann. Rev. As tron . As trophys . 4, 293.LEHNERT, . 1954 Astrophys . J . 119, 647.LEHNERT, . 1971 Nuclear Pwion, 11, 485.LEIGHTON, . B. 1969 Astrophys . J . 156, 1.LERCHE,. 1 9 7 1 a Astrophys . J . 166, 6 3 9 ,LERCHE,. 1971b Astrophys . J . 168, 123.LERCHE,. & Low, B . 4 . 1971 Astrophys . J . 168, 503.LEVY,E. H. 1 9 7 2 a Astrophys . J . 171, 621.LEVY,E. H. 1972b Astrophys . J . 171, 635.LILLEY,F. E. M. 1970 Proc. Ro y. Soc . A316, 153.LOPER,D. E. 1 9 7 0 a P h y s . F l u i d s, 13, 2999.

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648 H . K . MoffattLOPER,D. E. 1 9 70 b P h y s . E a r t h P l a n . I n t , 4, 129.LOPER,D. E . 1971 J . Flu id Mech . 50, 609.LOPER,D. E . 1973 Ph ys . Ear th P lan . In t . ( t o appear).LOPER,D. E. & BENTON, . R. 1970 J . F l u i d M e c h . 43, 785 .MALKUS,W. V. R. 1967 J . Flu id Mech . 28, 793 .MALKUS,W. V. R. 1971 Lectures in A p p l i e d M a t h . 14, 207.MOKENZIE,. F . 1973 J . FZuid M e c h . (to appear).MESTEL, L. 1965 Stel lar and Solar Magnet ic Fields (ed. R . Lust) 87. North-Holland,Amsterdam.MESTEL, . 1 9 68 a M o n . No t . R o y . A s t r . Soc. 138, 359.MESTEL,L. 1 9 68 3 M o n . No t . R o y . A s tr . Soc. 140, 177.MESTEL, . 1 9 71 a Na t u r e P h y s . S c i . 233, 149.MESTEL,L. 1971 b Qua rt. J . R o y . A s t r . Soc. 12, 402.MESTEL, . 1972 Cosmic Plas ma Phys ics (ed. K . Schindler), p. 203. Plenum.MESTEL, . & SELLEY, . S. 1970 M on . Not . Roy . As t r . Soc. 149, 197.MESTEL, . & TAKHAR, . S. 1972 M on . Not . Roy . As t r . Soc . 156, 419.MOFFATT, . K . 1968 J . Fluid Mech. 35, 117.MOFFATT, . K. 1970a J . F l u i d M e c h . 41, 435.MOFFATT,H. K. 1970b J . Fluid M e c h . 44, 705 .MOFFATT, . K . 1972 J . F l u i d M e c h . 53, 385.MOORE, . R. & WEISS,N. 0. 1972 Submitted to J . F l u i d M e c h .OSTRIKER,. P. & GW", J. E. 1969 Astrophys. J . 157, 1395.PARKER,E . N. 1955 Astrophys. J . 122, 293.PARKER,. N. 1970a Ann. Rev . As t ron. As t rophys . 8, 1.PARKER,E. N. 1970b Astrophys . J . 160, 383.PARKER, . N. 1971a Astrophys . J . 163, 255.PARKER,E. N. 1971b Astrophys . J . 164, 491.PRESTON,.W. 1970 Stellar Rotation (ed.A. Slettebak),p. 254. Reidel, Dordrecht-Holland.PRESTON,. W. 1971 Pub . As t r . Soc . Pac . 83, 571.RADLER,K.-H. 1969 M o w t s b . D t . A k a d . Wws. Berl in , 11, 194.RIKITAKE, . 1958 Proc. Ca mb . Ph i l . Soc. 54, 89.ROBERTS, .0 . 1970 Phi l . T ran s . Ro y . Soc. A266, 535.ROBERTS, . 0 . 1972 Phi l . T ran s . Roy . Soc. A271, 411.ROBERTS,. H. 1 96 8 P h i l . T r a n s . R o y . Soc. A 263, 93.ROBERTS,. H. 1971 a In T h e World Magnet ic Survey (ed.A. J . Zmudit), p. 123. IUGG Publ.ROBERTS, . H. 1971b I n Lectures in Appl ied Mathemat ics 14, 12 9 (ed. W. H . Reid).American Mathematical Society.ROBERTS,. H. 1972 Phi l . Tra ns . Roy . Soc. A 272, 663.ROBERTS,. H . & SOWARD, . 1972 Ann. R e v . F l u i d M e c h . 4, 117.ROBERTS, . H . & STIX,M . 1971 The turbulent dynamo: a translation of a series ofROBERTS, . H. & STIX,M. 1972 Astron. & As t rophys . 18, 453.RUDERMAN,. 1969 J . Phys ique , 30, C3152.RUDERMAN, . 1972 Ann. Rev . As t ron. As t rophys . 10, 427 .RUDRAIAH, . & VENEATACHALAPPA,. 1972 J . Fluid Mech. 52, 193.SAKURAI,., CLARK, . JR.& CLARK,P . A . 1971 J . F l u i d M e ch . 49, 753.SELLEY, . S. 1973 In preparation.SEVERNY, . B. 1971 Quart . J . Ro y . A s t ron . Soc . 12, 363.SIMON,G. W. & WEISS,N. 0. 1968 2. As t rophys . 59, 435.SKILES,D. D. 1972 Ph ys . Ea rth Planet . Interiors, 5, 90.SKUMANICH,. 1972 Astrophys. J . 171, 565.SOWARD,. M. 1972 Piail. Tr an s. R o y. Soc. A 272, 431.SPITZER,L. JR. 1958 Electromagnetic Processes in Cosmical Physics (ed. B.Lehnert),STARR,V. P. & GILMAN, . A. 1965 Astrophys. J . 141, 1119.STEENBECK,. & KRAUSE, . 1969 Astr . Nach r . 291, 49.STEENBECK,., KRAUSE,. & RADLER,K. H. 1966 2. Naturforsch. 21a, 369.

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Magnetohydrodynamic phenomena in rotat ing $uids 649STEWARTSON,. 1967 Proc . Roy. Soc. A 299, 173.STEWARTSON,. & RICKARD,J.A. 1969 J . Fluid Mech. 35, 759.STIX,M. 1971 Astrom. & Astrophys. 13, 203.STIX,M. 1972 Astron. & Astrophys. 20, 9.TAYLOR,.M. 1963 Proc . Roy. Soc. A 274, 274.VERONIS, . 1966 J . Fluid Mech. 26 , 49.WEBER,E. J. & DAMS,L. JR. 1967 Astrophya. J . 148, 217.WEDEMEYER,. H. 1964 J . Fluid Mech. 20, 383.WEISS,N. 0. 1971 Solar Magnetic Fields (ed. R. Howard), p. 602. Reidel, Dordrecht.WILCOX, . M. 1 9 7 1 Solar Magnetic Fields (ed. R. Howard), p. 744. Reidel, Dordrecht.WRIGHT, . A. E. 1969 Mon. Not. R. Astr. Soc . 146, 197.WRIGHT,G.A. E. 1970 Ph.D. dissertation, Manchester University.WRIOHT,G . A. E. 1973 Mon. Not. R o y . Astr. Soc. (in Press).

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