Prandtl number dependence of convection driven dynamos in rotating spherical fluid shells
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t J. Fluid Mech. (2005), vol. pp. (cid:13)
Prandtl number dependence of convectiondriven dynamos in rotating spherical fluidshells
By R. S I M I T E V
A N D
F. H. B U S S E
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany [email protected] (Received 1 June 2004 and in revised form 20 Jan 2005)
The value of the Prandtl number P exerts a strong influence on convection driven dy-namos in rotating spherical shells filled with electrically conducting fluids. Low Prandtlnumbers promote dynamo action through the shear provided by differential rotation,while the generation of magnetic fields is more difficult to sustain in high Prandtl num-ber fluids where higher values of the magnetic Prandtl number P m are required. Themagnetostrophic approximation often used in dynamo theory appears to be valid onlyfor relatively high values of P and P m . Dynamos with a minimum value of P m seemto be most readily realizable in the presence of convection columns at moderately lowvalues of P . The structure of the magnetic field varies strongly with P in that dynamoswith a strong axial dipole field are found for high values of P while the energy of thiscomponent is exceeded by that of the axisymmetric toroidal field and by that of thenon-axisymmetric components at low values of P . Some conclusions are discussed in re-lation to the problem of the generation of planetary magnetic fields by motions in theirelectrically conducting liquid cores.
1. Introduction
The problem of the generation of magnetic fields by motions of an electrically con-ducting fluid in rotating spherical shells is one of the fundamental problems of planetaryand astrophysical sciences. The increasing availability in recent years of computer ca-pacity has permitted large scale numerical simulations of this process, but the creationof realistic models for planetary and stellar dynamos has been hampered by a lack ofknowledge about appropriate external parameters. Since molecular values of materialproperties are usually not attainable in computer simulations because of the limited nu-merical resolution, eddy diffusivities must be invoked for comparisons with observations.Eddy diffusivities represent the effects of the unresolved scales of the turbulent velocityfield and it is often assumed for this reason that the eddy diffusivities for velocities,temperature and magnetic fields are identical. The effects of turbulence on the diffusionof vector and scalar quantities differ, however, and large differences in the correspondingmolecular diffusivities are likely to be reflected in the effective diffusivities caused by thefluctuating velocity field of the unresolved scales. In the case of the Earth’s core, forexample, the magnetic diffusivity is assumed to have a value of the order of 2 m /sec (Braginsky and Roberts, 1995) which exceeds the kinematic viscosity by a factor of theorder 10 . Numerical simulations are thus capable of resolving magnetic fields, but arefar from resolving velocity structures. However, since the largest unresolved scales, v , l ,of velocity and length yield values for an eddy viscosity ν e ≈ v · l of the order 1 m /sec R. Simitev and F. H. Busse
PSfrag replacements zx y r θϕ Ω d Figure 1.
Geometrical configuration of the problem. A part of the outer spherical surface isremoved to expose the interior of the shell to which the conducting fluid is confined. or less, it is reasonable to assume magnetic Prandtl numbers less than unity. Since theconcept of eddy diffusivities is simplistic there have not been many theoretical deriva-tions for ratios of turbulent diffusivities which enter the dimensionless equations for thenumerical simulations. Some theoretical considerations in the astrophysical context canbe found in the paper by Eschrich and R¨udiger (1983). Values for an effective Prandtlnumber can eventually be derived from experiments on turbulent convection (see, forexample, Ahlers and Xu, 2001). Additional complications arise in planetary and astro-physical applications through anisotropies introduced by the effects of rotation and thepresence of large scale magnetic fields. While anisotropic eddy diffusivity tensors willbe useful for more realistic models, for the purpose of the present paper we prefer thesimplicity of scalar diffusivities.After introducing the basic equations and the method of their numerical solution insection 2 we shall consider the most important properties of convection without magneticfield as a function of the Prandtl number in section 3. Since a number of earlier papers onthis topic have been published this section can be kept relatively short. In section 4 theonset of convection driven dynamos in fluids with different Prandtl numbers is described.Energy aspects are considered in section 5 and the validity of the magnetostrophic ap-proximation is discussed in section 6. The influences of various boundary conditions areconsidered in section 7 and a concluding discussion is given in the final section 8.
2. Mathematical formulation of the problem and methods of solution
We consider a rotating spherical fluid shell as shown in figure 1. We assume that astatic state exists with the temperature distribution T S = T − βd r /
2. Here rd is thelength of the position vector with respect to the center of the sphere. The gravity field isgiven by g = − dγ r . In addition to d , the time d /ν , the temperature ν /γαd and themagnetic flux density ν ( µ̺ ) / /d are used as scales for the dimensionless description ofthe problem where ν denotes the kinematic viscosity of the fluid, κ its thermal diffusivity, ̺ its density and µ is its magnetic permeability. The equations of motion for the velocity randtl number dependence of convective dynamos u , the heat equation for the deviation Θ from the static temperature distribution,and the equation of induction for the magnetic flux density B are thus given by ∂ t u + u · ∇ u + τ k × u = −∇ π + Θ r + ∇ u + B · ∇ B , (2.1 a ) ∇ · u = 0 , (2.1 b ) P ( ∂ t Θ + u · ∇ Θ) = R r · u + ∇ Θ , (2.1 c ) ∇ · B = 0 , (2.1 d ) ∇ B = P m ( ∂ t B + u · ∇ B − B · ∇ u ) , (2.1 e )where ∂ t denotes the partial derivative with respect to time t and where all terms inthe equation of motion that can be written as gradients have been combined into ∇ π .The Boussinesq approximation has been assumed in that the density ̺ is regarded asconstant except in the gravity term where its temperature dependence given by α ≡− ( d ̺/ d T ) /̺ = const is taken into account. The Rayleigh number R , the Coriolis number τ , the Prandtl number P and the magnetic Prandtl number P m are defined by R = αγβd νκ , τ = 2Ω d ν , P = νκ , P m = νλ , (2.2)where λ is the magnetic diffusivity. Because the velocity field u as well as the magneticflux density B are solenoidal vector fields, the general representation in terms of poloidaland toroidal components can be used u = ∇ × ( ∇ v × r ) + ∇ w × r , (2.3 a ) B = ∇ × ( ∇ h × r ) + ∇ g × r . (2.3 b )By multiplying the (curl) and the curl of equation (2.1 a ) by r we obtain two equationsfor v and w [( ∇ − ∂ t ) L + τ ∂ ϕ ] ∇ v + τ Q w − L Θ = − r · ∇ × [ ∇ × ( u · ∇ u − B · ∇ B )] , (2.4 a )[( ∇ − ∂ t ) L + τ ∂ ϕ ] w − τ Q v = r · ∇ × ( u · ∇ u − B · ∇ B ) , (2.4 b )where ∂ ϕ denotes the partial derivative with respect to the angle ϕ of a spherical systemof coordinates r, θ, ϕ and where the operators L and Q are defined by L ≡ − r ∇ + ∂ r ( r ∂ r ) , Q ≡ r cos θ ∇ − ( L + r∂ r )(cos θ∂ r − r − sin θ∂ θ ) . The heat equation for the dimensionless deviation Θ from the static temperature distri-bution can be written in the form ∇ Θ + R L v = P ( ∂ t + u · ∇ )Θ , (2.5)and the equations for h and g are obtained through the multiplication of equation (2.1 e )and of its curl by r ∇ L h = P m [ ∂ t L h − r · ∇ × ( u × B )] , (2.6 a ) ∇ L g = P m [ ∂ t L g − r · ∇ × ( ∇ × ( u × B ))] . (2.6 b )We assume stress-free boundaries with fixed temperatures and use the value 0.4 for theradius ratio η = r i /r o , v = ∂ rr v = ∂ r ( w/r ) = Θ = 0 at r = r i ≡ / r = r o ≡ / . (2.7)For the magnetic field electrically insulating boundaries are assumed such that thepoloidal function h must be matched to the function h ( e ) which describes the poten-68 R. Simitev and F. H. Busse
Figure 2.
Equatorial streamlines r∂ ϕ v = const. in the case P = 0 . τ = 10 and R = 1 . × . × , 4 × , 9 × (from left to right). The two halves of last plot show convection inthe minimum and in the maximum of a relaxation cycle. tial fields outside the fluid shell g = h − h ( e ) = ∂ r ( h − h ( e ) ) = 0 at r = r i ≡ / r = r o ≡ / . (2.8)But computations for the case of an inner boundary with no-slip conditions and anelectrical conductivity equal to that of the fluid have also been done. The numericalintegration of equations (2.4),(2.5) and (2.6) together with boundary conditions (2.7)and (2.8) proceeds with the pseudo-spectral method as described by Tilgner and Busse(1997) which is based on an expansion of all dependent variables in spherical harmonicsfor the θ, ϕ -dependences, i.e. v = X l,m V ml ( r, t ) P ml (cos θ ) exp { imϕ } (2.9)and analogous expressions for the other variables, w, Θ , h and g . P ml denotes the asso-ciated Legendre functions. For the r -dependence expansions in Chebychev polynomialsare used. For further details see also Busse et al. (1998) or Grote et al. (1999). For thecomputations to be reported in the following a minimum of 33 collocation points in theradial direction and spherical harmonics up to the order 64 have been used. But in manycases the resolution has been increased to 49 collocation points and spherical harmonicsup to the order 96 or 128.
3. Convection in rotating spherical shells
Three different types of convection can be distinguished in rotating spherical shells.Predominantly convection occurs in the form of rolls aligned with the axis of rotationwhich exhibit properties of thermal Rossby waves in that they are drifting in the progradeazimuthal direction. They are confined to the region outside the virtual surface of thetangent cylinder which touches the inner boundary at its equator. The dynamics of theseconvection columns, as they are sometimes called, is intimately connected for Prandtlnumbers of the order unity or less with the differential rotation that is generated bytheir Reynolds stresses. Above their onset as a m -periodic pattern in the azimuthaldirection the convection columns experience with increasing Rayleigh number transitionsto amplitude and shape vacillations before they become spatio-temporally chaotic inthe dimensions perpendicular to the axis while retaining their nearly perfect alignmentwith the rotation vector. In this regime of beginning turbulence coherent processes suchas localized convection and relaxation oscillations are realized through the interactionof the differential rotation and the convection columns. A graphical display of thesestages of convection is shown in figure 2 in terms of the streamlines of the convection randtl number dependence of convective dynamos Figure 3.
Equatorial streamlines r∂ ϕ v = const. in the case P = 20, τ = 5 × and R = 1 . × , 1 . × , 2 . × , 1 . × (from left to right). Figure 4.
Equatorial streamlines r∂ ϕ v = const. in the case P = 0 . τ = 10 and R = 4 × ,6 × , 8 × , 10 (from left to right). columns intersected by the equatorial plane. For a more detailed description we referto Grote and Busse (2001) or the review of Busse (2002a). As the Reynolds stressesdecrease with increasing P these coherent processes disappear. Figure 3 indicates thatthe spiralling nature of the convection columns also diminishes with increasing Rayleighnumber. A small non-geostrophic differential rotation persist driven as a ”thermal wind”by temperature gradients caused by the lateral inhomogeneity of the convective heattransport.A second distinct form of convection are the equatorially attached cells which representmodified inertial modes and become the preferred form of convection at sufficiently lowvalues of P . They were first found by Zhang and Busse (1987) and analytical descrip-tions in terms of perturbed inertial oscillations have been given by Zhang (1994, 1995)and Busse and Simitev (2004a). The equatorially attached cells do not develop strongReynolds stresses and they are thus less subject to the disruptive effects of the shear ofa differential rotation. But the continuity of the convective heat transport requires thatthe equatorially attached convection occurs in conjunction with the columnar convectioncloser to the inner boundary of the shell. This effect is visible in the form of secondaryextrema of the streamlines far from the boundary in figure 4 where the evolution of theconvection flows in the chaotic regime is illustrated. Note that the strong attachment tothe outer equatorial boundary persists even at the highest value of R used in this figure.For further details see Simitev and Busse (2003).A third form of convection is realized in the polar regions of the shell which comprisethe two fluid domains inside the tangent cylinder. Since gravity and rotation vectors arenearly parallel in these regions (unless large values of η are used) convection resemblesthe kind realized in a horizontal layer heated from below and rotating about a verticalaxis. A tendency towards an alignment of convection rolls with the North-South direction(Busse and Cuong, 1977) can be noticed, but this property is superseded by instabilities70 R. Simitev and F. H. Busse
Figure 5.
Polar convection in the cases, P = 0 . τ = 5 × , R = 2 × ; P = 0 . τ = 3 × , R = 7 . × ; P = 1, τ = 10 , R = 1 . × and P = 20, τ = 5 × , R = 10 (from left to right). The plots in the upper row show lines of constant u r at the surface r = r i + 0 . u ϕ in the left half, of Θ in the upper right quarter and of the streamlines r sin θ∂ θ v in the lower right quarter, all in the meridional plane. of the K¨uppers-Lortz type and by interactions with turbulent convection outside thetangent cylinder. The onset of convection in the polar regions generally occurs at Rayleighnumbers considerably above the critical values R c for onset of convection outside thetangent cylinder. Except for the case of very low Prandtl numbers the differential rotationin the polar regions is usually oriented in the direction opposite to that of rotationand thus tends to facilitate polar convection by reducing the rotational constraint. Thepossibility exists, however, that at sufficiently low values of P and high values of τ finiteamplitude convection in the polar regions may precede the onset of convection in otherregions. The patterns of polar convection and other properties discussed in this sectionare illuminated by the display of convection at different values of P in figure 5.Quantitative aspects of convection are described by figure 6 where the averages overspace and time of the kinetic energy densities of the various components of the convectionflow are shown as a function of R . The energy densities are defined by E p = 12 h| ∇ × ( ∇ ¯ v × r ) | i , E t = 12 h| ∇ ¯ w × r | i , (3.1 a )ˇ E p = 12 h| ∇ × ( ∇ ˇ v × r ) | i , ˇ E t = 12 h| ∇ ˇ w × r | i , (3.1 b )where the angular brackets indicate the average over the fluid shell and ¯ v refers to theazimuthally averaged component of v , while ˇ v is defined by ˇ v = v − ¯ v . The Nusseltnumber at the inner spherical boundary N u i is also shown in figure 6. It is defined by N u i = 1 − Pr i dΘd r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r i (3.2)where the double bar indicates the average over the spherical surface. The rapid growth randtl number dependence of convective dynamos P S f r ag r e p l a c e m e n t s − − . . . . . (a) ( b ) (c) ( d ) ( R − R c ) /R c P S f r ag r e p l a c e m e n t s − − . .
11 100.1 8151 . . ( a ) (b) ( c ) (d) ( R − R c ) /R c Figure 6.
Time-averaged energy densities E t (squares), ˇ E p (plus-signs), ˇ E t (crosses) and Nusseltnumber Nu i (filled triangles, right ordinates) as functions of R/R c for (a) P = 0 . τ = 10 , (b) P = 0 . τ = 3 × , (c) P = 1, τ = 10 , and (d) P = 20, τ = 5 × . The values of R c are 283000, 235000, 190000 and 146000 in the cases (a) , (b) , (c) and (d) respectively. The meanpoloidal energy component E p has been omitted since it is by factor 10 –10 smaller than theother components. with R of E t corresponding to the energy of differential rotation is remarkable in the cases (a) , (b) , and (c) of figure 6. Only for higher values of P as in case (d) does E t never exceedthe energies of the fluctuating components of motion. Another remarkable feature of theplots is the rapid rise of the Nusselt number N u i after the onset of amplitude vacillationsand the transition to a chaotic time dependence. The steadily drifting convection columnsor the equatorially-attached cells do not transport heat very well because of the mismatchbetween the geometry of the boundaries and the flow structure. Because this mismatchdoes not occur in the case of polar convection, the heat transport of the latter mayeasily exceed that of convection outside the tangent cylinder at higher values of R . Thedependence of the kinetic energy densities on R in case (a) of figure 6 are somewhatunusual in that after a rapid growth near onset a regime of saturation follows and agrowth of the kinetic energies with R begins again only when R has reached nearlytwice its critical value. The behavior is caused by the competition of several modes withdifferent azimuthal wavenumbers m as is described in more detail by Simitev and Busse(2003). The ways in which properties of convection are reflected in the generated magneticfields will be discussed in the following sections.72 R. Simitev and F. H. Busse P S f r ag r e p l a c e m e n t s Rm P m Figure 7.
Magnetic Reynolds numbers Rm for the onset of dynamo action as a function of P m in the cases P = 0 . τ = 10 (stars), P = 0 . τ = 10 (crosses), P = 0 . τ = 10 (circles), P = 1, τ = 3 × (triangles up), P = 1, τ = 10 (squares) and P = 5, τ = 5 × (diamonds)and P = 10, τ = 5 × (triangles down). The empty symbols in the cases with P > .
4. The onset of dynamo action for different Prandtl numbers
The onset of convection driven dynamos in the case P = 1 has been explored in nu-merous papers in the past. We refer to the work of Kageyama and Sato (1997), Kida andKitauchi (1998), Ishihara and Kida (2002), Busse et al. (1998), Christensen et al. (1999),Olson et al. (1999), Katayama et al. (1999), Takahashi et al. (2003), Grote et al. (2000),Grote and Busse (2001). In particular the results of the latter two papers can be com-pared well with the results presented in the following since only the Prandtl number differsamong the parameters of the computations. Since one of the goals of the numerical sim-ulations is the search for dynamos with low values of the magnetic Prandtl number P m ,the amplitude of convection must be sufficiently high such that the magnetic Reynoldsnumber defined by Rm = (2 E ) / P m reaches a value of the order 10 . Here E refers tothe average density of the kinetic energy of convection, i.e. E = E p + E t + ˇ E p + ˇ E t .In figure 7 values of Rm for cases of sustained dynamo action and for cases of decayingmagnetic fields have been indicated for several values of P . No special effort has beenmade to determine the minimum value Rm min of Rm for dynamo action. Since in allcases convection is chaotic and since dynamos are typically subcritical, i.e. the boundarybetween sustenance of a dynamo and its decay occurs at finite amplitudes of the magneticfield, Rm min is not a well defined quantity. Moreover, the value Rm as defined abovewill decrease after the onset of dynamo action since typically the energy of differentialrotation is strongly reduced by the action of the Lorentz force.It is evident from figure 7 that the goal of low values P m is most readily reached atlow Prandtl numbers. At high values of P the minimum value of the magnetic Reynoldsnumbers for dynamos decreases slightly, but P m must increase in proportion to P in orderto achieve a finite ratio P/P m in the limit of infinite Prandtl number as shown by Zhangand Busse (1990). The tendency of low Prandtl numbers to lower the minimum values of P m required for dynamos is limited, however, by the transition to inertial convection for P ( τ /τ ) − where τ is a quantity of the order 10 (Ardes et al., 1997). It thus has notbeen possible to obtain dynamos with values of P m below 0 . randtl number dependence of convective dynamos PSfrag replacements ⊙⊙⊙⊙⊙⊙ ⊙⊙⊙ ⊙⊙ ⊙⊙ ⊙⊙⊙⊙
N NNNN NNN NNNNNNN NNNN NNNNNNN⋆⋆ ⋆⋆ ⋆⋆⋆⋆⋆⋆ ⋆⋆ △△ (cid:4) (cid:4) (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) (cid:4) (cid:4) PP P m R × − R × − Figure 8.
Convection driven dynamos as a function of R , P and P m for τ = 5 × . The symbolsindicate chaotic dipolar ( N ), hemispherical ( △ ), quadrupolar ( ⋆ ), mixed ( (cid:4) ) and decayingdynamos ( ⊙ ). than 0 .
05. Inertial convection is evidently less conducive to dynamo action than columnarconvection and higher values of R m are required for dynamos at P = 0 .
025 and P = 0 . P m dynamos thus appears to be an increasing τ with an accompanying reduction of P . Of course, the numerical resolution will have to be increased at the same time.A particular feature that is evident from figure 7 (and also figure 9) is that high valuesof R may prevent dynamo action. In the case P = 0 . τ = 10 it is found that only afinite interval of convection energy permits dynamo action since high values of Rm leadto the expulsion of magnetic flux from the convection eddies which is detrimental for thegeneration of magnetic fields. The same effect can also be noticed for higher values ofthe Prandtl number as demonstrated in the case of P m = P = 10 of figure 8.A more complete overview of dynamos at values of P and P m above unity is shownin figure 8 where the structure of the magnetic field is also indicated. As long as theproperties of convection are nearly symmetric with respect to the equatorial plane threetypes of dynamos can be distinguished: Quadrupolar dynamos are characterized by amagnetic field that exhibits the same symmetry with respect to the equatorial plane asthe convection velocity field, while the opposite symmetry characterizes dipolar dynamos.Hemispherical dynamos correspond to the superposition of a quadrupolar and a dipolarmagnetic field of nearly the same form, but with opposite signs in the northern andthe southern hemispheres such that the magnetic field nearly vanishes in one of thehemispheres (Grote and Busse, 2000). Once the symmetry of convection with respect74 R. Simitev and F. H. Busse P S f r ag r e p l a c e m e n t s P m R Figure 9.
Convection driven dynamos as a function of the Rayleigh number R and the magneticPrandtl number P m for P = 0 . τ = 10 (empty symbols) and τ = 3 × (filled symbols). Thesymbols indicate chaotic dipolar (squares), hemispherical (triangles), mixed (stars) and decayingdynamos (circles). to the equatorial plane is impaired significantly, – as happens, for example, at higherRayleigh numbers through the onset of polar convection –, the structures of dynamoscan no longer be clearly distinguished and we speak of “mixed” dynamos. These dynamoscontinue to exhibit an alignment with the axis of rotation. Dynamos with a dominantequatorial dipole, for instance, have not been found.For a fixed value of P quadrupolar dynamos are obtained for sufficiently low valuesof P m which give way to hemispherical and dipolar dynamos with increasing P m as hasalso been observed in the case P = 1 (Grote et al. , 2000). But since dynamos for lowvalues of P m disappear with increasing P quadrupolar and hemispherical dynamos canno longer be obtained as P exceeds a value of the order 5 in the case of τ = 5 × .Quadrupolar dynamos thus appear to be restricted to Prandtl numbers of the order unityand moderate values of τ . When τ is increased from 10 to 3 × quadrupolar dynamosdo no longer seem to be accessible according to the results of Grote et al. (2001, see alsoBusse, 2002a). The same effect occurs as P is lowered as indicated in figure 9 for the case P = 0 . τ = 10 . Here the lowest value of P m corresponds to a hemispherical dynamo.It should be noted that the critical value R c of the Rayleigh number increases nearlyin proportion to τ P for P < τ can thus be reached forlower values of P . The degree of chaos increases, however, with decreasing P because ofthe short thermal time scale. Nevertheless coherent structures are also found such as therelaxation oscillations apparent in the time record shown in figure 10. Here the energydensities on the various components of the magnetic field are shown which are definedin analogy to the kinetic energy densities (3.1), M p = 12 h| ∇ × ( ∇ ¯ h × r ) | i , M t = 12 h| ∇ ¯ g × r | i , (4.1 a )ˇ M p = 12 h| ∇ × ( ∇ ˇ h × r ) | i , ˇ M t = 12 h| ∇ ˇ g × r | i . (4.1 b )In contrast to the relaxation oscillations of non-magnetic convection mentioned in con-nection with figure 2 the oscillations of figure 10 have a much shorter period and originatefrom the oscillations of the hemispherical magnetic field as is evident from the comparable randtl number dependence of convective dynamos P S f r ag r e p l a c e m e n t s M dip M quad E t × Figure 10.
Time series of energy densities of a hemispherical dynamo in the case P = 0 . τ = 10 , R = 6 × , P m = 0 .
11. The upper and middle panels show energy densities of dipolarand quadrupolar components of the magnetic field, while the lower panel shows energy densitiesof the velocity field. The mean toroidal components are represented by solid lines, the fluctuatingtoroidal - by dotted lines, the mean poloidal - by dot-dashed lines and the fluctuating poloidalby dashed lines. periods. Since the strength of the magnetic field varies more with the phase of the cycleat low values of P than at higher ones the action of the Lorentz force on the differentialrotation also exhibits significant variations. This in turn affects the amplitude of con-vection and its dynamo action and leads to the cyclic feedback process evident in figure10. We note that the computational results displayed in figure 10 were obtained with aresolution of 33 collocation points in the radial direction and with spherical harmonicsof the order 96. Results obtained with 41 collocation points and spherical harmonics upto the order 108 are rather similar and the time averaged energies differ insignificantly.Finally, we like to point out some typical differences between high and low Prandtlnumber dipolar dynamos which are exhibited by the plots of figure 11. At low valuesof P the quadrupolar component is usually quite noticeable leading to a shift of themagnetic equator away from the geometric one. Often this shift alternates in time ina nearly periodic fashion. In high Prandtl number dynamos the mean magnetic field isnearly steady and the quadrupolar component is negligible if the Rayleigh number is nottoo high. This is evident in the lower plots of figure 11 even though the onset of polarconvection with its attendant asymmetry with respect to the equatorial plane can alreadybe noticed. A characteristic feature of high P dynamos are the strong zonal magneticflux tubes in the polar regions which are mainly caused by the thermal wind shear an76 R. Simitev and F. H. Busse
Figure 11.
Non-oscillatory chaotic dipolar dynamos in the cases P = 0 . τ = 10 , R = 2 × , P m = 3 (upper row) and P = 200, τ = 5 × , R = 10 , P m = 80 (lower row). The left columnshows meridional isolines of B ϕ (left half) and of r sin θ∂ θ h (right half). The middle columnshows lines B r = const. at r = r . The right column shows lines u r = const. at r = r i + 0 . example of which can be seen in the case P = 20 of figure 5. These flux tubes are alsovery evident in the dynamo of Glatzmaier and Roberts (1995). While the differentialrotation changes a lot with increasing Prandtl number, the structure of the convectioncolumns hardly varies as is also evident from figure 11. The rate of drift in the azimuthaldirection decreases, of course, with increasing P .
5. Energetics of dynamos
It is of interest to compare the various interactions between velocity and magneticfield components which sustain dynamo action against Ohmic dissipation. Zhang andBusse (1989) have listed a total of 31 interaction integrals which contribute to the energybalances obtained from equations (2.6) for the mean and fluctuating components of thepoloidal and toroidal magnetic fields. Fortunately only a few of the 31 terms contributesignificantly and we list here only the most important ones,¯ p ≡ (ˇ v ˇ g ¯ h ) , ¯ p ≡ ( ˇ w ˇ h ¯ h ) , ¯ p ≡ (ˇ v ˇ h ¯ h ) , (5.1 a )¯ t ≡ ( ¯ w ¯ h ¯ g ) , ¯ t ≡ ( ˇ w ˇ h ¯ g ) , ¯ t ≡ (ˇ v ˇ g ¯ g ) , (5.1 b )ˇ p ≡ (ˇ v ¯ g ˇ h ) , ˇ p ≡ (ˇ v ˇ g ˇ h ) , ˇ p ≡ (ˇ v ˇ h ˇ h ) , (5.1 c )ˇ t ≡ ( ˇ w ¯ g ˇ g ) , ˇ t ≡ ( ˇ w ˇ g ˇ g ) , ˇ t ≡ ( ˇ w ¯ h ˇ g ) , (5.1 d )where the first two letters inside the brackets indicate which of the interactions betweenvelocity and magnetic field components on the right hand sides of equations (2.6) coun-teract the Ohmic dissipation of the magnetic field component indicated by the last letterinside the brackets. In the case of chaotic dynamos these integrals tend to fluctuatewildly and may change their signs. The particular ones listed above have been plotted asa function of time in figure 12 together with the corresponding average Ohmic dissipa-tion density for three representative cases. It is remarkable that the differential rotation randtl number dependence of convective dynamos PSfrag replacements 0.34 0.390.400.00 0.03 0.070.060.08 17 19 22010 200 1010-1.5-0.51.5 0000 11 22 30.01.02.0-0.10.0 0.1-0.20.00.200.20.4 0.00.00.20.20.40.40.0 .02.04 .00.00.00 .04.04.08 pt ˇ p ˇ t ttt × × Figure 12.
Lorentz terms (the thin solid, dotted, and dashed lines indicate the first, second,and third terms, respectively, in each of the lines of (5.1)) and Ohmic dissipations (thick dottedshaded lines) in the cases P = 0 . τ = 10 , R = 6 × , P m = 0 .
15 (left column), P = 0 . τ = 10 , R = 3 × , P m = 3 (middle column), P = 200, τ = 5 × , R = 10 , P m = 80 (rightcolumn). contributes most of the sustenance of the mean toroidal field not only in the case oflow values of P and P m , but also in the high Prandtl number case. The mean thermalwind is obviously sufficient to generate the mean toroidal field through the distortionand stretching of the mean poloidal field. This ω -effect is thus usually found to dominatethe dynamo process in the case of the zonal field. Only in particularly chaotic dynamoscharacterized by a rather weak mean poloidal field is the ω -effect not effective as in thecase P = 0 . , P m = 3 of figure 12. It is remarkable that the first and second terms ofdefinitions (5.1a) are strongly anticorrelated in the low Prandtl number cases of figure12. Because of their near cancellations the axisymmetric poloidal field is rather weak inlow P dynamos.The α -effect in which the fluctuating magnetic field is generated through the interactionbetween fluctuating velocity field and mean magnetic field plays a far less important role78 R. Simitev and F. H. Busse P S f r ag r e p l a c e m e n t s − E x , M x P = 1 P = 3 P = 5 P = 15 P = P = P = P = V x , O x P m P S f r ag r e p l a c e m e n t s − E x , M x P = P = P = P = P = 1 P = 3 P = 5 P = 15 V x , O x P m Figure 13.
Kinetic E x and magnetic M x energy densities (left) and viscous V x and Ohmic O x dissipations (right) as functions of P m for convection driven dynamos for τ = 5 × , R = 10 and Prandtl number as indicated in the boxes. The highest values of the Elsasser number forthe cases P = 1, 3, 5, 15 are Λ = 3 .
02, 0 .
31, 0 .
37 and 0 .
49, respectively. The components X p , X t , ˇ X p , ˇ X t (where X = E , M , V , O ) are represented by circles, squares, plus-signs and crosses,respectively. Kinetic energy densities and viscous dissipations are shown with light symbols,magnetic energy densities and Ohmic dissipations are shown with heavy symbols. than the ω -effect. The fluctuating components of the magnetic field are usually generatedthrough interactions of fluctuating parts of velocity and magnetic fields except in the caseof high Prandtl numbers where the mean poloidal field typically dominates.One of the least understood questions of convection driven dynamos is the equilibrationof magnetic energy. While in the astrophysical context an equipartition between kineticand magnetic energy is often favored, an Elsasser number Λ of the order unity is regardedby geophysicists as a good estimate of the magnetic energy generated by dynamos inplanetary cores. In the following we shall use the definitionΛ = 2 M P m τ (5.2)for the Elsasser number where M denotes the magnetic energy density averaged over thefluid shell and in time, i.e. M = M p + M t + ˇ M p + ˇ M t . Already Chandrasekhar (1961)established that for Λ = 1 a minimum is found of the critical Rayleigh number for onsetof convection in a horizontal layer heated from below, penetrated by a homogeneousvertical magnetic field and rotating about a vertical axis.In figures 13 and 14 averaged kinetic and magnetic energy densities and viscous andohmic dissipation densities have been plotted for typical parameter sets of the problem.The Elsasser number has been computed for the largest magnetic energy M in each ofthe boxes and is given in the figure captions. It is evident that Λ rarely exceeds unityand when it does a physical reason of a high value of Λ is not obvious. In particular, inthe cases of figure 13 where only P m is varied it is hard to understand why the Elsasser randtl number dependence of convective dynamos P S f r ag r e p l a c e m e n t s − E x , M x V x , O x
23 10 101420 2020 20 203050 6 613 3 5 7
60 80 (a) (b) (c) (d) (e) (f) (g) E x , M x V x , O x R × − P S f r ag r e p l a c e m e n t s − E x , M x V x , O x
23 10 101420 2020 20 203050 6 613 3 5 7
60 80 (a) (b) (c) (d) (e) (f) (g) E x , M x V x , O x R × − Figure 14.
Kinetic E x and magnetic M x energy densities (upper row) and viscous V x andOhmic O x dissipations (lower row) as functions of R for convection driven dynamos in the cases (a) P = 0 . τ = 10 , P m = 0 .
2, Λ = 0 . (b) P = 0 . τ = 10 , P m = 3, Λ = 3 . (c) P = 0 . τ = 3 × , P m = 1, Λ = 1 . (d) P = 1, τ = 3 × , P m = 2, Λ = 0 .
26; (e) P = 1, τ = 5 × , P m = 1, Λ = 0 .
93; (f) P = 5, τ = 5 × , P m = 10, Λ = 0 .
68 and (g) P = 10, τ = 5 × , P m = 10, Λ = 0 .
12. The components X p , X t , ˇ X p , ˇ X t (where X = E , M , V , O )are represented by circles, squares, plus-signs and crosses, respectively. Kinetic energy densitiesand viscous dissipations are shown with light symbols, magnetic energy densities and Ohmicdissipations are shown with heavy symbols. In case (b) results for a convection driven dynamoas well as for convection without magnetic field have been plotted in the case R = 2 × . Thevalues of Λ represent the highest values for each of the boxes. R. Simitev and F. H. Busse
Figure 15.
Convection driven dynamo for P = 0 . τ = 10 , R = 6 × and P m = 0 .
5. Thefirst plot shows lines of constant u ϕ in the left half and streamlines, r sin θ∂ θ v = const. in theright half. The second plot shows streamlines in the equatorial plane, r∂ ϕ v = const.. The lefthalf of the third plot shows lines of constant zonal magnetic field B ϕ and the right half showsmeridional field lines, r sin θ∂ θ h = const.. The last plot shows lines of constant B r at the surface r = r o + 0 . number increases more strongly than P m itself. Another way of looking at the problem isto determine the highest value of Λ as a function of the Rayleigh number. A systematicstudy of the variation of magnetic energies with increasing R has been undertaken byGrote and Busse (2001) in the case P = P m = 1, τ = 5 × . This study has beenextended since that time in order to obtain reliable time averages, see figure 11 of Busseand Simitev (2004b). Here the case of maximal M is attained for R = 1 . × in part (e) of figure 14 and corresponds to a value of about unity for Λ. But this result may beaccidental as is suggested by the magnetic fields in the cases of dynamos for low valuesof P and P m . Here again we find in the case (a) of figure 14 that magnetic energy hasreached a value close to its maximum as a function of R , but the corresponding value0 .
38 of Λ is significantly lower. A much higher value of Λ is obtained, on the other hand,when P m = 0 . P m = 3 even though M may still increase with a furtherincrease in R . We thus conclude that the criterion Λ ≈ P m is actually weaker than suggested by the definition (5.2) of Λ.There are a number of other features exhibited by the diagrams of figures 13 and 14.While the energy densities of the fluctuating components of the magnetic field usuallyexceed those of the mean components this situation is reversed at sufficiently high Prandtlnumbers for the poloidal part of the field as is evident from the last columns of figure13 as well as of figure 14. This changeover occurs rather suddenly at about P = 8 for τ = 5 × as can be noticed in the comparison of cases (f) and (g) of figure 14. A closerinspection shows that the replacement of the geostrophic differential rotation by thethermal wind and the accompanying growth of the polar zonal flux tubes is responsiblefor this change in the dynamo process. It should be noticed that this changeover is notevident in the plots of the dissipation densities since the latter are always much lower forthe mean components than for the fluctuating components.Another property that changes with increasing Prandtl number is the fact that for P of the order unity or less the energy of the mean toroidal field exceeds that of themean poloidal field except for some high Rayleigh number cases where polar convectionbegins to dominate. For dipolar dynamos obtained for P = 0 . τ = 10 with Rayleighnumbers of the order 10 and 0 . P m . . For example, in thecase R = 10 , P m = 0 . M p = 146 and M t = 2 . × are found. The structure of velocity and magnetic fields in this case is randtl number dependence of convective dynamos P S f r ag r e p l a c e m e n t s × RR E t ˇ E t Nu i Figure 16.
Time-averaged kinetic energy densities and Nusselt number Nu i of non-magneticconvection (thick lines) and of quadrupolar (diamonds), mixed (stars), hemispherical (triangles)and chaotic dipolar dynamos (squares). Both non-magnetic convection and dynamo solutionsare in the case P = 1 and τ = 10 . Values of P m of the dynamo cases decrease from 10 to 0 . R increases. The densities E t , ˇ E t and Nu i are shown in the left, middle and right panel,respectively. apparent from figure 15. The axisymmetric components of the magnetic field are almoststeady in time. But the differential rotation in the polar regions may change its sign.At the particular time of figure 15 the zonal flows near the north- and south-poles haveopposite signs. The fluctuating component of convection is dominated by the m = 1-modeand exhibits the attachment to the equatorial boundary as must be expected for inertialconvection according to the discussion in section 3. For Prandtl numbers of the order 5and higher, on the other hand, the energy of the mean poloidal field always exceeds thatof the mean toroidal field, since the differential rotation which creates the latter is notstrong enough. But this changeover is rather gradual and depends more on the Rayleighnumber than the other changeover discussed above.Since in some of the cases of figure 14 the kinetic energies of convection in the absenceof a magnetic field are shown, the effect of the magnetic field on convection becomesevident. In particular in the cases with P of the order unity or less the energy of themean toroidal velocity field is strongly reduced by the dynamo action in that the Lorentzforce leads to a braking of the differential rotation as has been discussed by Grote andBusse (2001). On the other hand the fluctuating components of convection are enhancedsince the magnetic field prevents the onset of relaxation oscillations in which convectionoccurs only intermittently as has been discussed in connection with figure 2. This effectis clearly demonstrated in case (b) of figure 14 where the results for convection withand without magnetic field are compared for R = 2 × . The same features are alsorevealed in figure 16 where a more general comparison between convection with andwithout magnetic fields is shown. This figure also demonstrates that the symmetry ofthe magnetic field does not play a major role in the considerations of energetic aspectsof dynamos.For Prandtl numbers larger than unity the effect of the magnetic field is much reducedwhich is evident from the fact that the increasing magnetic energy for increasing P m does not change the energy of the fluctuating components of convection as can be seenin figure 13. Only the differential rotation is reduced through the strengthening of theLorentz force. The diminished influence of magnetic field on convection is also evident82 R. Simitev and F. H. Busse from the property that typically viscous dissipation exceeds Ohmic dissipation by far forvalues of P of the order 5 or higher, while the two sinks of energy are more comparableat low values of P .
6. Validity of the magnetostrophic approximation
Because convection driven dynamos in rotating systems depend on a rather large num-ber of parameters it is desirable to eliminate one or more parameters through reductionsof the basic equations. Among the nonlinear advection terms the momentum advectionterm appears to be most expendable since it does not seem to be essential for convectiondriven dynamos. In this way the magnetostrophic approximation is obtained in whichthe acceleration of fluid particles is neglected in comparison to the Coriolis force andthe Lorentz force. In general this approximation can be easily obtained through the re-placement of the viscous time scale d /ν by the time scale d / ( κ − γ λ γ ) with 0 γ p ρµκ − γ λ γ ν/d is used as scale of the magnetic field the basic dimensionless equationsof motion, equation of induction and the heat equation can be written in the form( P − γ P γm ) − ( ∂ t u + u · ∇ u ) + τ k × u = −∇ π + Θ r + ∇ u + ( ∇ × B ) × B (6.1 a )( P m /P ) γ ( ∂ t B + u · ∇ B − B · ∇ u ) = ∇ B (6.1 b )( P/P m ) − γ ( ∂ t Θ + u · ∇ Θ) = R u · r + ∇ Θ (6.1 c )where k is the unit vector parallel to the axis of rotation. From the form of equations(6.1) it is clear that the magnetostrophic approximation should certainly be valid in thelimit P − γ P γm −→ ∞ . In the following we shall focus on the case γ = 0. In figure 17the energy densities have been plotted for fixed values of R, τ and of κ/λ . Since thefluctuating poloidal energy density ˇ E p always amounts to about 50 percent of the cor-responding toroidal one it has not been plotted. It can be seen that the kinetic energiestend to become independent of P with increasing P in accordance with the magne-tostrophic assumption. The energy density E t representing the differential rotation isthe only exception as expected. Little indication of an approach towards the validity ofthe magnetostrophic approximation is found, however, when the magnetic energy densi-ties are considered. As has already been mentioned the dynamo process is rather sensitiveto the presence of the differential rotation and much higher values of P may be neededbefore the magnetostrophic regime is approached. It is remarkable to see the distinctminimum of magnetic energies near P = 8 which corresponds to change in the structureof the magnetic field as has been mentioned. The transition from a geostrophic differentialrotation to a thermal wind one appears to be mainly responsible for this feature.Results obtained on the basis of the magnetostrophic approximation in the case γ = 0are by definition independent of P . In particular the ratio between the magnetic energyand kinetic energy will be proportional to P (Glatzmaier and Roberts, 1995) as is evidentfrom the different scales used for the velocity and for B / √ ρµ in equation (6.1 a ). Theratio between Ohmic and viscous dissipation would be independent of P and woulddepend only on κ/λ . This latter parameter seems to be even more important than themagnetic Prandtl number for convection driven dynamos. At least for Prandtl numberof the order unity or less the condition 1 . P m /P appears to apply for the minimumvalue P m for which dynamo action can be obtained as is indicated by a comparison ofdynamos obtained for P = 1 (see figure 1 of Grote et al. , 2002) and for P = 0 . γ = 0 may be reached for γ = 1. In this randtl number dependence of convective dynamos P S f r ag r e p l a c e m e n t s M p P M t P ˇ M p P ˇ M t P (a) (b) (c)(d) (e) (f) P P P − Figure 17.
Kinetic energy densities E p (in (a) ), ˇ E t (in (b) ) and E t (in (c) ), all multipliedby P as functions of P in the case τ = 5 × . The dynamos corresponding to κ/λ = 1, R = 5 × are indicated by solid squares, to κ/λ = 1, R = 6 × by solid triangles, to κ/λ = 1, R = 8 × by solid circles, to κ/λ = 2, R = 6 × by crosses, to κ/λ = 2, R = 10 by plus-signs, to κ/λ = 5, R = 6 × by empty squares, to κ/λ = 0 . R = 10 by emptytriangles and to non-magnetic convection with R = 10 by empty circles. The second row showsthe corresponding magnetic energy densities multiplied by P . case the energy densities of figure 17 can be plotted as a function of P m . For the cases κ/λ = 1 the plots remain unchanged with P m replacing P at the abscissa. For othervalues of κ/λ the respective data would just be shifted. Since no particular new insightappears to be evident we have not included such a figure. The general conclusion to bedrawn is that the magnetostrophic approximation seems to become valid only for fairlyhigh values of the parameter P − γ P γm with 0 γ
1. It should also be rememberedthat properties such as inertial convection and geostrophic differential rotation generatedby Reynolds stresses disappear when the magnetostrophic approximation is used. Kuangand Bloxham (1996) have introduced for this reason a compromise in that the magne-tostrophic approximation is applied only to the non-axisymmetric part of the equationsof motion. But their main result that the ratio of magnetic to kinetic energy reaches 10 for P = 1 is far removed from properties of other convection driven spherical dynamosdescribed in the literature.
7. Effects of boundary conditions on dynamo solutions
Because published solutions of convection driven dynamos in rotating spherical shellsare based on a variety of parameter values and satisfy different mechanical, thermaland magnetic boundary conditions, a systematic comparison between various models isdifficult to conduct. An attempt to investigate the differences between numerical models84
R. Simitev and F. H. Busse
Table 1.
Time-averaged global properties of dynamos with various velocity and magneticboundary conditions as follows. A : stress-free and insulating, B : no-slip and insulating, C :no-slip and a finitely-conducting inner core and D : stress-free and a perfectly-conducting innercore. The predominant symmetry type is indicated with “D” if dipolar, “Q” if quadrupolar and“–” if the dynamo is decaying. In the case B in left part of the table results for both a dynamosolution and a non-magnetic convection case are given. P = 0 . τ = 10 , R = 4 × , P m = 0 . P = 5, τ = 5 × , R = 8 × , P m = 3 A B B C D A B C D
Type D D – D Q Q – – Q E p .286 × .112 × .939 × .108 × .299 × .157 .611 .523 .147 E t .599 × .647 × .757 × .807 × .764 × .533 × .105 × .106 × .528 × ˇ E p .142 × .115 × .221 × .121 × .138 × .574 × .614 × .611 × .566 × ˇ E t .336 × .257 × .540 × .273 × .247 × .119 × .983 × .952 × .119 × M p .129 × .392 × – .355 × .768 × .487 ×
10 – – .466 × M t .133 × .879 × – .716 × .538 × .325 ×
10 – – .336 × M p .136 × .169 × – .147 × .148 × .818 ×
10 – – .837 × M t .307 × .345 × – .327 × .448 × .109 × – – .110 × V p .643 × .104 × .107 × .122 × .462 × .151 × .111 × .686 × .140 × V t .584 × .366 × .277 × .348 × .709 × .182 × .342 × .245 × .179 × ˇ V p .133 × .179 × .448 × .191 × .125 × .414 × .524 × .528 × .406 × ˇ V t .229 × .354 × .821 × .372 × .171 × .455 × .580 × .568 × .450 × O p .626 × .129 × – .120 × .874 × .544 × – – .520 × O t .148 × .944 × – .890 × .222 × .359 × – – .372 × ˇ O p .125 × .138 × – .133 × .106 × .468 × – – .488 × ˇ O t .253 × .268 × – .276 × .362 × .600 × – – .587 × Λ .707 .995 – .902 .740 .032 – – .032 Rm
164 138 – 142 152 64 55 55 64 Nu i based on various assumptions has most recently been made by Kutzner and Christensen(2000, 2002) who consider thermal and compositional driving with internal or externaldistribution of energy sources and a variety of thermal boundary conditions. Here wecomplement their studies with a few remarks on the possible effects of various velocityand magnetic boundary conditions. For this purpose we have selected two typical dynamosolutions. The dynamo at parameter values P = 0 . τ = 10 , R = 4 × , P m = 0 . P = 5, τ = 5 × , R = 8 × , P m = 3has a quadrupolar one. The results for the standard boundary conditions (2.7) and (2.8)used in this paper are listed under A . We have repeated the simulations of these twocases for three of models with different combinations of boundary conditions and presentthe results in table 1. For type B no-slip and insulating conditions have been adopted,for type C we have chosen no-slip conditions and a finitely-conducting inner core andfinally type D has stress-free velocity boundary conditions and a perfectly-conductinginner core. Thermal conditions have remained unchanged in that temperatures are fixedat the boundaries in all four models. Snapshots of the spatial structure of the magneticfield are shown in figures 18 and 19.The replacement of the stress-free condition by the no-slip condition at the inner andouter boundaries leads to considerable changes in the structure of the velocity field whichare even more dramatic in the absence of a magnetic field. The differential rotation is randtl number dependence of convective dynamos Figure 18.
Effects of various boundary conditions on dynamo solutions for P = 0 . τ = 10 , R = 4 × , P m = 0 . A , B , C (from left to right) of table 1. The lefthalves of each plot show lines of constant zonal magnetic field B ϕ and the right halves showmeridional field lines, r sin θ∂ θ h = const., at particular moments in time. Figure 19.
Same as figure 18 but for the quadrupolar cases D and A and D for the parametervalues P = 5, τ = 5 × , R = 8 × , P m = 3. strongly inhibited by the Ekman layers which tend to oppose any deviation from rigidbody rotation. The circulation induced by the Ekman layers, on the other hand, promotesthe heat transport and and is responsible for the fact that the energy component E p istypically much larger in the case of no-slip boundaries than in the case of stress-freeones. But the main effect of the rigid boundaries is the suppression of the destructionof convection columns by the shearing action of the differential rotation. Relaxationoscillations which are predominant for Prandtl numbers of the order unity or less inthe presence of stress-free conditions have not yet been found when no-slip conditionsare used. It must be expected, however, that the properties of convection with rigidboundaries will approach those of convection with free boundaries as the Coriolis number τ increases and the influence of the Ekman layer decreases.The effects caused by the differences in the velocity boundary conditions are reducedin the presence of a magnetic field generated by the dynamo process. The differentialrotation is reduced by the action of the Lorentz force for both types of boundary con-ditions. But while the energy of the fluctuating components of convection is amplifiedby the presence of the magnetic field this effect usually does not happen in the case ofno-slip boundaries as is evident from case B of table 1. On the contrary, the magneticdegrees of freedom compete with kinetic ones for the same source of energy. Nevertheless,in spite of the lower kinetic energies in case B in comparison to those of case A , case B still exhibits a higher convective heat transport as is evident from the Nusselt number.The comparison of convection driven dynamos in case of P = 5 as presented by theright side of table 1 has been less successful since it was not possible to obtain dynamosin the cases B and D even after the Rayleigh number has been increased up to twicethe indicated value. Except for the axisymmetric components of the velocities the kinetic86 R. Simitev and F. H. Busse energies of the cases A and B are surprisingly close which indicates that the effect ofno-slip boundaries on convection is similar to that of a dynamo generated magnetic field.The closeness of the energy results of the cases A and D for both values of P and of cases B and C for P = 0 . D a quadrupolar dynamo is realized instead of the dipolar one in the standard case A .Even more remarkable is the property that this quadrupolar dynamo does not oscillate intime as has been found for all quadrupolar dynamos with the boundary conditions (2.8).From a general point of view it must be expected that for turbulent dynamos the innercore can not exert a strong influence on the dynamo process since the highly fluctuatingmagnetic field will not penetrate far into the inner core nor is it likely that strong fluxeswill accumulate there since they are subject to Ohmic decay. Wicht (2002) has arrivedat similar conclusions.
8. Concluding remarks
In this paper the analysis of convection driven dynamos has focused on the Prandtlnumber dependence while other parameters such as R and τ have not been varied asmuch, but instead have been kept at reasonably high values which are numerically acces-sible with adequate resolution in space and in time. In the case of the Rayleigh numberthe numerical limitations are not felt as strongly as in the case of τ since dynamos oftenseem to disappear or at least exhibit decreasing energies of the axisymmetric componentsof their magnetic fields as R is increased beyond an optimal value. More restrictive arethe limits on the Coriolis number τ which have prevented so far the attainment of dy-namos dominated by Ohmic dissipation. While the latter exceeds the viscous dissipationin some of the low Prandtl number dynamos, sufficient viscous friction always seems tobe required to obtain numerically well-behaved solutions. Ohmic dissipation also helpsin this respect and permits solutions at values of R and τ which are not accessible atthe same numerical resolution without magnetic field. But Ohmic dissipation does notnecessarily prevent tangential discontinuities of the velocity field which are possible inthe absence of viscous friction.A primary goal of dynamo simulations are results which can be compared quantita-tively with properties of the geomagnetic field and its variations in time. The geodynamoappears to exhibit features which are similar to those of convection driven dynamos withlarge values of P as well as to those with P of the order of unity. The fact that the mag-netic energy exceeds the kinetic energy in the Earth’s core by a factor of the order 10 together with the property that the axisymmetric poloidal component of the geomagneticfield dominates over the non-axisymmetric components suggest that the geodynamo re-sembles a high- P dynamo. On the other hand, the strong variations of the amplitude ofthe magnetic field on the magnetic diffusion timescale together with the torsional oscil-lations which manifest themselves as “jerks” on a much shorter timescale (Bloxham etal. , 2002) indicate a relationship to dynamos with P . P m become possible. But they could alsoindicate that the two sources of buoyancy apparently present in the Earth’s core, – thethermal one characterized by P < P ≫ randtl number dependence of convective dynamos REFERENCESAhlers, G., and Xu, X.
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