Dynamics of convection and dynamos in rotating spherical fluid shells

Fluid Dynamics Research 28 (2001) 349–368

Dynamics of convection and dynamos in rotating spherical "uidshells

E. Grote, F.H. Busse ∗

Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany

Received 1 August 2000; accepted 9 January 2001

Abstract

Numerical simulations of thermal convection in rapidly rotating spherical "uid shells have been carried out with andwithout magnetic /elds generated by the dynamo process. Relaxation oscillations and localized convection activity representcoherent phenomena of turbulent convection in the absence of magnetic /elds. In the presence of the latter, the coherentstructures are destroyed and the heat transport is enhanced. With increasing Rayleigh number the magnetic energy tendsto saturate and the magnetic /eld assumes an increasingly /lamentary structure. c© 2001 Published by The Japan Societyof Fluid Mechanics and Elsevier Science B.V. All rights reserved.

1. Introduction

It is generally believed that the fact that most rotating celestial bodies with a "uid interior exhibit amagnetic /eld is intimately connected with the property that magnetic /elds facilitate the convectiveheat transport in the presence of rotation. Already in the case of a plane horizontal "uid layerheated from below and rotating about a vertical axis, it can easily be seen that the imposition ofa vertical magnetic /eld can counteract the stabilizing in"uence of the Coriolis force and thus cansubstantially reduce the critical Rayleigh number for the onset of convection (Chandrasekhar, 1961).Similar e;ects of imposed magnetic /elds have been found for convection in other rotating layers(Eltayeb and Roberts, 1970; Eltayeb, 1972) or in the case of centrifugally driven convection in thecylindrical annulus (Busse, 1976; Busse and Finocchi, 1993). Most relevant for geophysical andastrophysical applications is the case of convection in a rotating sphere with a given azimuthalmagnetic /eld which has been studied by Fearn (1979a, b) and more recently by Zhang (1995). Forthe close connection of this problem with the problem of the cylindrical annulus see the review ofBusse (1983a).

∗ Corresponding author. Tel.: +49-921553329; fax: +49-921552999.E-mail address: [email protected] (F.H. Busse).

0169-5983/01/$20.00 c© 2001 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V.All rights reserved.PII: S0 1 6 9 - 5 9 8 3 ( 0 1 ) 0 0 0 0 4 - 1

350 E. Grote, F.H. Busse / Fluid Dynamics Research 28 (2001) 349–368

The magnetic /eld acting in celestial bodies or in the Earth’s core is not an imposed one. Itis generated through the dynamo process by the same convective motions which it is supposed topromote against the inhibiting in"uence of the Coriolis force. This property changes the nature ofthe interaction between convection "ow and magnetic /eld and leads to a far more intricate situationthan in the case of a prescribed magnetic /eld.

Several groups have studied numerically dynamos in rotating spherical shells driven by convection"ows at high Taylor numbers (Busse et al., 1998; Christensen et al., 1999; Grote et al., 1999, 2000;Olson et al., 1999 and others), based on the same or a rather similar model as will be assumed inthe present paper. For a review we refer to Busse (2000). The properties of non-magnetic convectionand the changes that occur after the onset of dynamo action have received relatively little attentionand therefore will be at the focus of the present paper. The ways in which magnetic /elds saturatewith increasing amplitude of convection are also not yet fully understood and will be discussed inSection 5.

After an introduction of the basic mathematical model in Section 2 some typical dynamicalfeatures of convection in rotating spheres will be explored in Section 3. The changes that areintroduced when the convection "ow acts as a dynamo are studied in Section 4. We shall seethat the generated magnetic /eld still tends to facilitate the convective transport of energy, atleast for suGciently high rotation rates. But the main e;ect of the magnetic /eld occurs indi-rectly through the braking of the di;erential rotation which is quite di;erent from the case of animposed azimuthal magnetic /eld. After a discussion in Section 5 of the evolution of dynamoswith increasing Rayleigh number an outlook on future research will be given in the concludingremarks.

2. Mathematical formulation of the problem

We follow the standard formulation of the problem used in earlier work by the authors (Busseet al., 1998; Grote et al., 1999, 2000) by considering a spherical "uid shell of thickness d which isrotating about a /xed axis with the constant angular velocity �. We assume that a static state existswith the temperature distribution TS = T0 − �d2r2=2 and that the gravity /eld is given by g =−�dr,where r is the position vector with respect to the center of the sphere and r is its length measuredin units of d. In addition to the length d, the time d2=�, the temperature �d2�= and the magnetic"ux density �(%)1=2=d are used as scales for the dimensionless description of the problem, where �denotes the kinematic viscosity of the "uid, its thermal di;usivity, is the magnetic permeabilityand % its density. The latter is assumed to be constant except in the gravity term where its temperaturedependence given by � ≡ (d%=dT )=%=const: is taken into account, i.e. the Boussinesq approximationis used.

Since the velocity /eld u as well as the magnetic "ux density B are solenoidal vector /elds, wecan use the general representation in terms of poloidal and toroidal components,

u =∇× (∇v × r) +∇w × r; (1a)

B =∇× (∇h × r) +∇g × r: (1b)

E. Grote, F.H. Busse / Fluid Dynamics Research 28 (2001) 349–368 351

By multiplying the (curl)2 and the curl of the Navier–Stokes equations in the rotating system by rwe obtain two equations for v and w,

[(∇2 − @t)L2 + �@’]∇2v + �Qw − RL2� =−r · ∇ × [∇× (u · ∇u − B · ∇B)]; (2a)

[(∇2 − @t)L2 + �@’]w − �Qv = r · ∇ × (u · ∇u − B · ∇B); (2b)

where @t and @’ denote the partial derivatives with respect to time t and with respect to the angle’ of a spherical system of coordinates r; �; ’ and where the operators L2 and Q are de/ned by

L2 ≡ −r2∇2 + @r(r2@r);

Q ≡ r cos �∇2 − (L2 + r@r)(cos �@r − r−1 sin �@�):

The heat equation for the dimensionless deviation � from the static temperature distribution can bewritten in the form

∇2� + L2v = P(@t + u · ∇)� (2c)

and the equations for h and g are obtained through the multiplication of the equation of magneticinduction and of its curl by r,

∇2L2h = Pm[@tL2h − r · ∇ × (u × B)]; (2d)

∇2L2g = Pm[@tL2g − r · ∇ × (∇× (u × B))]: (2e)

The Rayleigh number R, the Coriolis parameter �, the Prandtl number P and the magnetic Prandtlnumber Pm are de/ned by

R =��d6

�; � =

2�d2

�; P =

�; Pm =

��; (3)

where � is the magnetic di;usivity. We assume stress-free boundaries with /xed temperatures,

v = @2rrv = @r(w=r) = � = 0 at r = ri ≡ �=(1− �) and at r = ro = (1− �)−1: (4a)

where � is the radius ratio, �= ri=ro. Throughout this paper only the case �=0:4 will be considered.For the magnetic /eld electrically insulating boundaries are used such that the poloidal function hmust be matched to the function h(e) which describes the potential /elds outside the "uid shell

g = h − h(e) = @r(h − h(e)) = 0 at r = ri and r = ro: (4b)

The assumption of an insulating inner core is not very realistic in view of planetary or stellarapplications. It has been introduced here for reasons of simplicity. The numerical integration of Eqs.(2) together with boundary conditions (4) proceeds with the pseudo-spectral method as described byTilgner and Busse (1997) which is based on an expansion of all dependent variables in sphericalharmonics for the �; ’-dependences, i.e.

v =∑

l;m

V ml (r; t)P

ml (cos �) exp{im’} (5)

and analogous expressions for the other variables, w;�; h and g. Pml denotes the associated Legendre

functions. For the r-dependence expansions in Chebychev polynomials are used. For further detailssee also Busse et al. (1998).

For most of the computations reported in the following, 33 collocation points in the radial directionand spherical harmonics up to the order 64 have been used.


3. Finite amplitude convection in rotating spherical shells

Numerous papers have been published on the subject of convection driven by thermal or chemicalbuoyancy in rotating spherical shells. Reviews of some of this work have been given by Tilgneret al. (1997) and by Zhang and Busse (1998). While large parts of the parameter space have beenexplored in the published literature there are still some qualitatively new phenomena which have notbeen noticed before and to which we wish to draw attention in the following.

Except for very low values of the rotation parameter � (Geiger and Busse, 1981), convection ina rotating sphere sets in the form of motions which are symmetric with respect to the equatorialplane. For Prandtl numbers of the order unity or larger the convection assumes the form of periodiccolumns which drift in the prograde azimuthal direction. In this stage convection "ows are con/nedto the region outside the cylindrical surface touching the inner core at its equator (also known as“tangent cylinder”) if � is suGciently high. As the Rayleigh number increases beyond the onsetvarious scenarios involving changes in the azimuthal wave number usually occur (see, for example,Sun et al., 1993a; Ardes et al., 1997), and a preference for vacillations of the amplitude can beobserved, especially if the Prandtl number is of the order unity or less. As the variations in time ofthe amplitude increase with increasing R, a transition to a chaotic state usually occurs. The columnsare no longer regularly spaced around the circumference and the time dependence is described by abroad spectrum of frequencies. But the symmetry about the equatorial plane is still well preserved forconvection outside the tangent cylinder, especially if � is suGciently large. Even at Rayleigh numbersof 100 times the critical value this property can still be noticed (Sun et al., 1993b; Christensen, 2001).Signi/cant components of convection which are antisymmetric with respect to the equatorial planedevelop inside the tangent cylinder when the Rayleigh number becomes high enough that the criticalvalue for the onset of convection in the polar regions is exceeded. This is due to the fact thatthere is no communication between the convection "ows in the two polar regions in contrast tothe convection columns which obey the Proudman–Taylor condition in /rst approximation. In thefollowing we wish to consider the evolution of convection with increasing R for two examples inmore detail.

We start by considering the case P=1; �=104 for which the critical Rayleigh number Rc is closeto 1:9 × 105 where a drifting pattern of columnar convection with wave number m0 = 10 will beobtained. This steadily drifting form of convection is characterized by constant azimuthally averagedproperties such as the Nusselt number Nui at the inner boundary which is de/ned by

Nui − 1 =−Pri

@ PP�@r

∣∣∣∣∣∣r=ri

; (6)

where PP� indicates the average of � over a spherical surface.A transition to a truly time dependent state occurs at Rayleigh numbers slightly above 2:6× 105,

where the solution of vacillating convection bifurcates from the steady branch. The convection isstill strictly periodic in the azimuthal direction with the wave number m0 = 10, but its amplitudevaries periodically in time. A typical example of these amplitude vacillations is shown in Fig. 1. Atthe Rayleigh number of 3 × 105 used for this /gure, the time dependence has already experienceda doubling of the period as is evident from the time series of the kinetic energies plotted in Fig. 2.


Fig. 1. Vacillating convection in the case � = 104; R = 3 × 105; P = 1. The upper six plots are equidistant in time (leftto right) with Qt = 0:018 and cover one period of oscillation such that the sixth plot is identical with the /rst one. Eachplot displays lines of constant Pu’ in the left upper quarter, streamlines of the axisymmetric component of the velocity/eld in the meridional plane in the right upper quarter, and streamlines (r(@=@’)v = const:) in the equatorial plane. Thelower six plots display lines of ur = const: at the surface r = (ri + ro)=2 for the same instances of time. Solid (dashed)lines indicate positive (negative) values.

These energies are de/ned by

PEp = 12〈|∇ × (∇ Pv × r)|2〉; PEt = 1

2〈|∇ Pw × r|2〉; (7a)

REp = 12〈|∇ × (∇ Rv × r)|2〉; REt = 1

2〈|∇ Rw × r|2〉; (7b)

where the brackets 〈: : :〉 indicate the average over the spherical shell, Pv denotes the axisymmetriccomponent of v and Rv ≡ v − Pv denotes the azimuthally "uctuating component of v. The energy PEp


Fig. 2. Energies of convection as a function of time t in the case � = 104; P = 1. PEt (thick solid lines), REt (thin solidlines), REp (dotted lines) have been plotted for R = 2:8 × 105; 3 × 105; 3:5 × 105; 4 × 105; 7 × 105; 1:2 × 106 (from top tobottom). The energy PEp is several orders of magnitude smaller than the others and has not been plotted for this reason.

has not been plotted in Fig. 2 since it is usually smaller than the other kinetic energies by severalorders of magnitude. It is remarkable to see how the time average of PEt increases relative to theenergies of the non-axisymmetric components of motion which are directly driven by buoyancy. Asis evident from the plots of Pu’ in Fig. 1, PEt corresponds to a di;erential rotation which is nearlyindependent of the coordinate in the direction of the axis of rotation. As this di;erential rotationincreases it tends to shear o; the spiraling convection columns such that temporarily an inner ringand outer ring of convection columns result as is most clearly shown in the /fth plot of the upperpart of Fig. 1. In the following sixth plot the inner and outer columns have already reconnectedafter a jump of 360◦ in phase. As the shear of the di;erential rotation increases further, the strain onthe convection pattern no longer permits a time-periodic response and temporally chaotic convectionresults at a Rayleigh number of about 3:1× 105.

Up to this point the evolution of convection with increasing Rayleigh number parallels the scenariodescribed by Sun et al. (1993a) in the case � = 103. In the turbulent state which develops as R isincreased further not much regularity can be seen until Rayleigh numbers of the order 5 × 105


and higher are reached. At this point a new localized coherent convection structure develops. Thechaotic equilibrium between the di;erential rotation generated by the Reynolds stresses exerted bythe spiraling convection columns (Busse, 1983b) and destruction of convecting eddies caused by theshear /nally leads to a structure in which the convection columns occupy only a small fraction of thespherical shell, while the rest of the "uid shell is motionless except for the axisymmetric di;erentialrotation. This turbulent coherent structure is evident in the time series of plots shown in Fig. 3.While the convection columns drift through the structure in the prograde direction in the outer part,a retrograde motion of the eddies can be noticed near the inner boundary. The latter phenomenonis caused by the retrograde di;erential rotation which exceeds in absolute value the prograde driftvelocity of the convection columns. The opposite shift in phase between real and imaginary partsof V 8

8 (r; t) near the inner and outer boundary as shown in Fig. 4a clearly indicates the oppositepropagation of the columns.

The most interesting feature of the turbulent convection spot is its driving mechanism. In thepresence of convection the temperature near the inner boundary is lowered while it is increasedtowards the outer boundary as can be noticed in the plots of the lower half of Fig. 3. In theinterior of the "uid shell the temperature distribution has thus become more isothermal and thereforeless capable of providing buoyancy for driving "uid motion. As this temperature distribution isadvected in the prograde direction in the outer part and in the retrograde direction on the inner side,the convection columns decay and thermal conduction tends to reestablish the basic temperaturedistribution with � = 0. As this latter pro/le is advected into the convection spot on its oppositeside it can provide suGcient buoyancy to sustain the convecting eddies against the shearing actionof the di;erential rotation and the associated viscous dissipation. The convection eddies thus bene/tfrom the fact that the stabilizing in"uence of the change in the mean temperature pro/le is reducedin that it is distributed over a region much wider than the convectively active one. On a longer timescale the coherent structure of chaotic convection moves slowly in the retrograde direction as canalready be noticed in Fig. 3, but is more clearly evident in Fig. 4b where the time dependence ofthe m = 1-component of the poloidal "ow is shown.The localized convection structure persists up to Rayleigh numbers of the order 106. But beyond

this range the di;erential rotation has become so strong that even the concentrated convection activitycan no longer be sustained against the stresses exerted by the radial (in the cylindrical sense) shear.In addition to the spatial intermittency an intermittency in time becomes necessary in order topermit convection, as is evident from the time series for the highest value of R in Fig. 1 andfrom various cases shown in Fig. 5. In the absence of convection the di;erential rotation decayssince there are no Reynolds stresses to sustain it. As the shearing action of the di;erential rotationbecomes suGciently weak convection columns grow in amplitude. But as their Reynolds stressesregenerate the di;erential rotation, their amplitude quickly peaks and then decays as the shearingaction interrupts the convection "ows. It is surprising how nearly periodically this process repeatsitself even though every convection episode di;ers from the next one in detail. Fig. 6 shows thesudden growth and decay of convection through a time series of plots. It is remarkable to see thatthe onset of convection occurs in a nearly spatially periodic fashion near the boundaries and onlylater a more chaotic pattern of convection develops. From the isotherms, �= const:, it can be seenthat the available buoyancy is reduced with the growth of the amplitude of convection and thata nearly isothermal state is established in the interior which also contributes to the decay of thepoloidal kinetic energy.


Fig. 3. Localized turbulent convection in the case �=104; R=7× 105; P =1. The upper six plots describe a time serieswith equidistant steps (Qt = 0:00246, left to right) of the streamlines (r(@=@’)v = const:) in the equatorial plane. In thelower six plots isotherms, �= const:, in the equatorial plane are plotted for the same instances in time. The direction ofrotation is anticlockwise as the equatorial plane is viewed from the north side.

The period of these relaxation oscillations is primarily governed by the viscous decay of thedi;erential rotation. The value of about 0.1 was found for the period over a wide range of theparameters R; � and P. But the fraction of this period during which the convection bursts occurvaries considerably. It tends to decrease with increasing Rayleigh number and decreasing Prandtlnumber.


Fig. 4. (a) Variation in time of real (solid lines) and imaginary (dotted lines) parts of the coeGcient V 88 (r; t) (see expression

(5)) at the position close to the inner boundary r= ri (thick lines) and close to the outer boundary (thin lines). Note thatthe phase shift between real and imaginary parts is opposite in the two cases. (b) Variation in time of the real (solid)and imaginary (dotted) parts of the coeGcient V 1

1 (r; t) at r = 12 (ri + ro).

4. Convection driven dynamos

For Prandtl numbers of order unity or less it is found that convection "ows have become chaoticbefore dynamo action sets in unless the magnetic Prandtl number Pm is signi/cantly larger than 10.Since dynamos of geophysical or astrophysical interest are characterized by small values of Pm, agoal of the numerical dynamo simulations has to reach a value of Pm that is as small as possible.Within the limits imposed by the power of present day computers it has been found diGcult to reachvalues of Pm as low as 0.1. Since dynamos in rotating spherical shells require a magnetic Reynoldsnumber Rem of the order 100 based on an average amplitude of the velocity /eld, values of at leastPm = 10 are required for dynamos generated by steadily drifting or time periodic convection. Thiscan be seen from Fig. 1 if the de/nition

Rem =√2Pm( PEt + PEp + REt + REp)1=2 (8)

is used. Steady and time periodic dipolar dynamos have been obtained for Pm¿30 as described byGrote et al. (2000). Here we shall focus on values of Pm of order unity.

From the numerous simulations without the use of hyperdi;usivities that have been carried outby various groups (Christensen et al., 1999; Olson et al., 1999; Grote et al., 2000) for values of �around 104 or less it can be concluded that the Lorentz force has little e;ect on the structure of theconvection columns. But their time dependence and the ratio between the energies of axisymmetricand of non-axisymmetric components of the velocity /eld are strongly a;ected. This is strikinglydemonstrated in Fig. 7, where the dynamo has suddenly disappeared, as often happens in the case ofsubcritical chaotic dynamos near the lower boundary of the range in R of their existence. The steady


Fig. 5. Energies of convection as a function of time t in the case � = 1:5 × 104; P = 0:5. The energies PEt (thick solidlines), REt (thin solid lines) and REp (dotted lines) have been plotted for R= 4× 105; 5× 105; 6× 105; 7× 105; 8× 105; 106

(from top to bottom). PEp is not shown since it is smaller than the other energies by several orders of magnitude.

(in the statistical sense) turbulence changes rather rapidly into the state of relaxation oscillationsdiscussed in the preceding section. The time average of the energy PEt which measures the strengthof the di;erential rotation increases while all other kinetic energies show a slight decrease in theaverage. The average Nusselt number is considerably smaller for the relaxation oscillation than inthe presence of the magnetic /eld. A similar, but not quite as strong, increase in the Nusselt numbercan also be noticed when the non-magnetic states of localized convection as shown in Fig. 3 arecompared with dynamo states for the same values of R; � and P.

Although the level of turbulence seems to increase after dynamo action starts and the spectrumof wave numbers participating in "uctuating components of motion broadens, coherent processes arestill operating in the interaction between velocity and magnetic /elds. Predominant among them isthe propagation of dynamo waves from low to high latitudes. As an example we consider here ahemispherical dynamo as shown in Fig. 8. As this time sequence of plots shows, azimuthal "ux


Fig. 6. Time series of plots near the peak of the convection amplitude in a relaxation oscillation at R =6:5 × 105; � = 104; P = 0:5. The upper six plots show lines Pu’ = const. on the left half and azimuthally averagedisotherms, P�=const., on the right half. The lower six plots indicate streamlines, r(@v=@’)=const., in the equatorial plane.In both cases the six plots are equidistant in time (left to right) with Qt = 0:01. The second plot in the lower sequencewould look similar to the /rst plot if the contours had not been drawn at one tenth of their usual values in order todemonstrate the regular onset of convection near the inner boundary and also near a section of the outer boundary. Thesense of rotation is anticlockwise.

PB’ of alternating polarity is generated in the equatorial plane near the inner boundary and migratesradially outward. It thereby pushes "ux of opposite sign towards the outer boundary and towardshigher latitudes.

This dynamo wave is connected primarily with the axisymmetric components of the magnetic /eldand appears as a coherent oscillation over long periods of time in the m=0 coeGcients. An exampleof a time sequence of coeGcients is shown in Fig. 9 which also demonstrates the contrasting chaotic


Fig. 7. Onset of relaxation oscillations after the decay of dynamo action in the case �=1:5×104; R=1:2×106; P=Pm=0:5.The total magnetic energy density M = PMp + PMt + RMp + RMt (dashed line), the kinetic energies PEt (thick solid line), PEp

(thick dotted line), REt (thin solid line), REp (lower thin dotted line) and the Nusselt number Nui (upper dotted line, rightordinate) are plotted as a function of time. To make it visible, PEp has been multiplied by a factor of 100.

time dependence of the non-axisymmetric components of velocity and magnetic /elds. The particularcase of a hemispherical dynamo used in Fig. 9 is of interest because it changes from a southernhemispherical dynamo to a northern one as shown in Fig. 10. This transition is also evident in lasttime record of Fig. 9 where the coeGcients with l = 1 and 2 are anti-correlated in the beginningof the time record and become correlated towards its end. As the intermediate state between thetwo hemispherical dynamos, a quadrupolar dynamo is realized which also exhibits the dynamo wavemigrating from the equator towards the poles with a slightly lower period than in the hemisphericalcase.

It should be mentioned here that hemispherical dynamos are not quite as common as has beenassumed in an earlier paper (Grote et al., 2000). In that paper most computations were carriedout with the use of only even azimuthal wave numbers m in order to save computer time. Whilequadrupolar dynamos do not seem to be a;ected much by this restriction, hemispherical dynamosexhibiting the symmetry with respect to the rotation by 180◦ turn out to be rather unstable withrespect to disturbances with the wave number m = 1. The region of existence of hemisphericaldynamos in the parameter space is thus much reduced.

The oscillations corresponding to waves propagating to higher latitudes have also been foundin the case of predominantly dipolar magnetic /elds. The oscillation period usually decreases withincreasing magnetic Reynolds number.

5. Energy considerations for spherical dynamos

The strength of cosmical magnetic /elds is one of the most easily determinable parameters. Butits relationship to other parameters of planetary and stellar dynamos is not well understood. A


Fig. 8. An oscillatory hemispherical dynamo in the case � = 104; R = 5× 105; P = 1; Pm = 5. A sequence of /ve plotsequidistant in time (from top to bottom) with Qt = 0:09 is shown such that approximately a full period of the dynamooscillation is covered between the uppermost and lowermost plots. Lines of constant Br on the surface r = ro are shownin the left column. The left half of the plots in the middle column display lines of constant B’ while the right half showsthe meridional /eld lines of the axisymmetric poloidal component of B. In the plots of the right column streamlines(r(@v=@’) = const:) in the equatorial plane are shown. The sense of rotation is left to right in the /rst column andanticlockwise in the third column.

popular proposal for convection driven dynamos in rotating systems is that the Elsasser numberde/ned by

) =B2D

2�%�(9)


Fig. 9. The dependence on the time t of various coeGcients of a hemispherical dynamo for �=1:5× 104; R=1:05× 106;P =0:5; Pm = 0:8 are shown. The plots show real (solid line) and imaginary (dotted line) parts of the coeGcient V 1

1 (r; t)of the poloidal component of the velocity /eld (uppermost plot), the coeGcient G0

1(r; t) of the toroidal component ofthe magnetic /eld (second plot), the real parts of the coeGcients H 2

2 (r; t) (solid line) and H 23 (r; t) (dotted line) of the

poloidal component of the magnetic /eld (third plot) and the coeGcients H 01 (r; t) (solid line) and H 0

2 (r; t) (dotted line)of the poloidal component of the magnetic /eld in lowermost plot.

assumes a value of the order unity in dynamos (Eltayeb and Roberts, 1970). Here BD denotes theaverage dimensional magnetic "ux density. This criterion is certainly much too simple and cannotprovide more than a rough guide for planetary dynamos. The numerical simulations have not yetprovided simple insights into the processes that determine the equilibration of the magnetic energy.In Fig. 11, time averages of the various kinetic and magnetic energies and rates of dissipation havebeen plotted as a function of the Rayleigh number for given values of �; P and Pm. Averages havebeen taken over time spans of the order unity which usually is suGcient to obtain a statisticallysigni/cant value. Most of the computations on which Fig. 11 is based have been carried out bothwith the use of only even azimuthal wave numbers and with the use of the full set of azimuthalwave numbers m, as indicated in the /gure. While the energies and dissipations do not di;er muchin the cases where both types of computations are available, there is a signi/cant di;erence at lowRayleigh numbers. In the presence of the full set of azimuthal wave numbers persistent dynamoscould not be obtained for Rayleigh numbers signi/cantly below 5× 105. This property is caused bythe “m=1-instability” which is the cause of localized convection and as a consequence leads to thedecay of dynamo action.


Fig. 10. Transition from a southern hemispherical dynamo to a northern one in the same case as in Fig. 9. The left halfof the left plots shows lines of constant PB’, the right half indicates the meridional /eld lines of the axisymmetric poloidalcomponent of B. The right plots show lines of constant Br at the outer surface, r = ro. The plots correspond to the timest = 0:86; 1:11 and 1.61 (from top to bottom) in Fig. 9.

A tendency towards saturation of the magnetic energies primarily of the quadrupolar components isnoticeable while neither Ohmic nor viscous dissipation nor kinetic energies exhibit such a tendency.At low values of R, the velocity /eld is nearly perfectly symmetric with respect to the equatorialplane and a purely quadrupolar dynamo is realized. As the Rayleigh number of 5× 105 is exceeded,convection sets in the polar regions and components of convection which are asymmetric with respectto the equatorial plane begin to grow. At the same time a dipolar component of the magnetic /eldis generated as must be expected since the loss of the symmetry of the velocity /eld implies thecorresponding loss for the magnetic /eld. It is remarkable how strongly the dipolar component ofthe magnetic /eld evolves with growing R such that its energies exceed those of the quadrupolarcomponent at the highest value of R reached in these computations. Typically, the dipolar /elddominates in the polar regions while the magnetic /eld tends to retain its quadrupolar symmetry inthe equatorial region as can be seen in Fig. 12. In Fig. 11 it can also be noticed that the energiesof the axisymmetric components of the quadrupolar /eld actually tend to decrease with increasing R


Fig. 11. Kinetic energy densities of symmetric (upper left) and antisymmetric (upper right) components, magnetic energydensities of quadrupolar (middle left) and dipolar (middle right) components, and viscous (lower left) and Ohmic (lowerright) dissipation are plotted as a function of R for convection driven dynamos in the case � = 5 × 103; P = Pm = 1.Filled (open) symbols indicate toroidal (poloidal) components of the energies and dissipations, circles (squares) indicateaxisymmetric (non-axisymmetric) components. In the case of the dissipations the contributions have not been separatedwith respect to their equatorial symmetry. The values of R at the abscissa should be multiplied by 105. The scales of theordinates in the two lower plots must also be multiplied by the factor 105. Symbols connected by solid (dotted) linescorrespond to computations carried out with all azimuthal wave numbers m (with even m only).

when R has reached values of the order 106 and in the case of the dipolar part of the magnetic /eldthe growth of the axisymmetric components is also much less than that of the non-axisymmetriccomponents. This property is caused by the increasing "ux expulsion with growing R which tendsto cause a /lamentary structure of the magnetic /eld as shown in Fig. 12. The same e;ect isalso responsible for the continuing strong growth of the Ohmic dissipation. At the Rayleigh numberR=16×105 of Fig. 12, the magnetic /eld is already dominated by the dipolar components, a tendencywhich is growing with increasing R. The "ux expulsion and the resulting separation between velocityand magnetic /eld may be a reason why an Elsasser number of the order unity is not always a goodmeasure of the strength of dynamo magnetic /eld. The values of ) are usually much less than 1 inthe case of Fig. 11, but approach unity at the higher Rayleigh numbers.


Fig. 12. Dynamo solution for R = 1:6× 106; � = 5× 103; P = Pm = 1 at a particular instant of time. The upper left plotshows lines of constant Pu’ in the left half and streamlines (r(@v=@’)=const:) in the equatorial plane in the right half. Theupper right plot shows lines of constant ur on the surface r=(ri + ro)=2. The lower two plots show the same informationfor the magnetic /eld as the upper plots for the velocity /eld except that the lower right plot shows the surface r = ro.

The results shown in Fig. 11 can also be compared with those obtained by Christensen et al. (1999)who have carried out similar systematic studies of convection driven dynamos. The tendency towardsan increasingly /lamentary structure of the magnetic /eld with increasing R is also seen in theirresults. Owing to the choice of a di;erent basic temperature pro/le without any internal heating theirdynamos usually exhibit a dipolar symmetry while in the case of Fig. 11, the quadrupolar componentprevails at least for lower Rayleigh number (Kutzner and Christensen, 2000). This di;erence couldalso be responsible for the property that the solutions of Christensen et al. (1999) for values ofPm of the order unity or less are always characterized by a predominant poloidal component of themagnetic /eld, while the toroidal parts of the magnetic /eld exceed in amplitude the poloidal partsin the dynamos described here and in earlier work (Grote et al., 1999, 2000).

It is of interest to consider in more detail the terms which sustain the di;erent components ofthe magnetic /eld against Ohmic dissipation. Among the numerous possible interactions betweenvarious components of the velocity and the magnetic /eld—there are 31 altogether as can be seenfrom Table 1 of Zhang and Busse (1989)—we list here only those which play the most importantrole in the case of the dynamos described in Fig. 11:

Pp1 ≡ ( Rv Rg Ph); Pp2 ≡ ( Rw Rh Ph); (10a)


Fig. 13. The terms Pp1 (solid line), Pp2 (dotted line) and Pt1 (dashed line) in the upper plot and the terms Rp1 (solid), Rp2(dashed) and Rp3 (dotted) in the middle plot and the terms Rt1 (solid), Rt2 (dashed) and Rt3 (dotted) in the lower plot aredisplayed as a function of time in the case � = 5× 103; R = 14× 105; P = Pm = 1.

Pt1 ≡ ( Pw Ph Pg); (10b)

Rp1 ≡ ( Rv Pg Rh); Rp2 ≡ ( Rv Rg Rh); p3 ≡ ( Rv Ph Rh); (10c)

Rt1 ≡ ( Rw Pg Rg); Rt2 ≡ ( Rw Rg Rg); Rt3 ≡ ( Rw Ph Rg); (10d)

where the /rst two letters inside the brackets indicate which of the interactions between velocityand magnetic /eld components on the right-hand sides of Eqs. (2d) and (2e) counteracts the Ohmicdissipation of the magnetic /eld component indicated by the last letter inside the brackets. In the caseof chaotic dynamos these integrals "uctuate wildly and may change their signs. The particular oneslisted above are displayed as a function of time in Fig. 13 for the case of R=14× 105; �=5× 103.In terms of the axisymmetric components of the magnetic /eld the dynamo could be called an�-!-dynamo in the nomenclature of mean /eld electrodynamics. But the "uctuating components ofthe magnetic /elds are generated primarily through the interaction between "uctuating componentsof the magnetic and of the velocity /eld. Only at Rayleigh numbers below 106 do the contributionsRp1 and Rt1 assume a more dominant role.


6. Concluding remarks

Except in the case of unrealistically large magnetic Prandtl numbers convectively driven dynamosin rotating systems are turbulent. The degree of turbulence already present in the absence of amagnetic /eld is usually enhanced when Lorentz forces enter the dynamics and the presence of themagnetic /eld opens new degrees of freedom. Nevertheless, convection driven dynamos in rotat-ing spheres mostly exhibit the structure of the convection "ow and di;er from the correspondingnon-magnetic states primarily in the character of the time dependence. Through the braking action ofthe magnetic /eld and the di;erential rotation, the dynamo states overcome the phenomenon of relax-ation oscillation and also to a considerable extent the phenomenon of localized convection. Therebydynamos do indeed eliminate some of the constraints of rotation and contribute to an enhancementof the convective heat transport in rotating spheres.

Though it has been possible to reach fairly high values of the rotation parameter in the simulationsreported in this paper, the region of the accessible parameter space is still rather restricted. Viscousdissipation still exceeds Ohmic dissipation in the case of Fig. 11 and the dynamo process maychange qualitatively as other regions of the parameter space are explored. Dynamos at low Prandtlnumbers seem to be the most easily accessible cases for which Ohmic dissipation dominates overthe viscous one (Grote et al., 2000). Continuing e;orts are needed to attain smaller values of themagnetic Prandtl number which are necessary for extrapolations of the numerical results to dynamosrealized in planetary cores and in stars.

Acknowledgements

The numerical simulations reported in this paper have been generously supported by the StuttgartSupercomputing Center.

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Dynamics of convection and dynamos in rotating spherical fluid shells