Hydrodynamic Helicity in Boussinesq-Type Models of the Geodynamo M

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ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2006, Vol. 42, No. 6, pp. 449Ð459. © Pleiades Publishing, Inc., 2006. Original Russian Text © M.Yu. Reshetnyak, 2006, published in Fizika Zemli, 2006, No. 6, pp. 3Ð13.

Hydrodynamic Helicity in Boussinesq-Type Models of the Geodynamo M. Yu. Reshetnyak Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia Received October 17, 2005

Abstract—The generation of hydrodynamic helicity is considered in models of thermal convection of a rapidly rotating incompressible liquid (small Rossby numbers). It is shown that a system of cyclonic vortices arising in geodynamo models generates a large-scale hydrodynamic helicity. The spatial distribution of the helicity is ana- lyzed as a function of the intensity of heat sources in models with various boundary conditions for the velocity ﬁeld. PACS numbers: 91.25.Cw DOI: 10.1134/S1069351306060012

INTRODUCTION The violation itself of the reflection invariance in the rotating turbulence is usually associated with the Cori- The origination of a large-scale magnetic field in olis force action on eddies moving in a medium with a celestial bodies is related to magnetic convection pro- zero gradient of density [Parker, 1979]. Let the density cesses. It is known that the magnetic field can be gener- of the body decrease with its radius. Then, an ascending ated by both large-scale flows [Bullard and Gellman, eddy broadens, producing a velocity component 1954] and small-scale flows [Vainshtein, 1983]. In the directed along the eddy radius. Accordingly, the Corio- latter case, along with small-scale fluctuating magnetic lis force arises due to the radial velocity and general fields, large-scale fields arise [Vainshtein et al., 1980] (a rotation. This force makes the eddy rotate in the sense synergic process), which is closely connected with the opposite to the general rotation. In a similar way, a concept of the α-effect in the theory of mean fields descending eddy contracts, also giving rise to a radial [Krause and Rädler, 1980]. Averaging over one coordi- velocity, and the Coriolis force makes the eddy rotate in nate, e.g., over the longitude, and using an additional the same sense as the general rotation. This results in a generation term with α, the 3-D dynamo problem could nonzero correlation, be reduced in many cases to the 2-D problem, while χH 〈〉⋅ preserving the validity of the limitation imposed by the = v curlv , (1) Cowling theorem, which precludes the generation of an i.e., the average helicity of the flow (here, v is the tur- axisymmetric magnetic field by an axisymmetric flow. bulent velocity and the broken brackets mean averag- α ing). Correlation (1) determines the value of the The origin of the -effect consists in the fact that the α mirror symmetry is violated in a convective rotating -effect (the Moffat formula): body and, as a result, the number of, e.g., clockwise H H ατ= Ð χ /3, (2) rotating eddies in the northern hemisphere is systemat- ically greater than the number of anticlockwise rotating where τH is the characteristic time of correlation. Note ones [Krause and Rädler, 1980]. On the contrary, anti- that the helicity V á curlV is interesting in itself, without clockwise rotating eddies in the opposite hemisphere regard for the α-effect, because it is the integral of dominate over clockwise rotating ones. This violation motion, like the kinetic energy [Kurganskii, 1993; Frik, of the mirror symmetry, resulting from the averaging of 2003; Lesieur, 1997]. In mechanics, the original of χH the Maxwell equations over turbulent pulsations, gives is the squared angular momentum [Dolzhanskii, 2001]. rise to a component of the average magnetic field B par- It is remarkable that the pseudoscalar value χH can be allel to the average electric current J = αB (whereas the estimated from symmetry considerations because a magnetic field is usually perpendicular to the current). pseudoscalar can be constructed from the density gra- It is this generally small parallel component of the mag- dient and angular rotation velocity (the so-called netic field that is capable of closing the self-excitation Krause formula, see for details [Reshetnyak and circuit of the magnetic field in the Faraday phenome- Sokolov, 2003]). This traditional interpretation of the non of electromagnetic induction. α-effect is, however, of limited applicability. First of

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450 RESHETNYAK all, the averaged flow V often cannot be separated from Ð1 ∂V EPr ------+ ()V ⋅ ∇ V the turbulent pulsations v in applied problems.1 The ∂t (3) α relationship between the helicity and -effect indicated = curlBB× Ð ∇P Ð 1 × V ++RaTr1 E∆V above has such a simple form only in the case of locally z r homogeneous and isotropic turbulence. Finally, the ∂ T ()⋅ ∇ ()∆ medium in which the magnetic field is generated is ------+ V TT+ 0 = T. ∂t often incompressible, so that the validity of the idea of expanding and compressing eddies is limited. It is The dimensionless numbers of Prandtl, Ekman, Ray- worth noting that, after the α-effect has been estab- ν ν lished, the variability of density of the medium is, as a leigh, and Roberts are written as Pr = --- , E = ------, κ Ω 2 rule, neglected in dynamo models. 2 L α δ η g0 TL ν The limitations noted above are particularly charac- Ra = ------Ωκ , and q = ---κ , where is the kinematic vis- teristic of two very important cases of planetary and 2 cosity coefficient, α is the coefficient of volume expan- laboratory dynamos. In the laboratory experiment car- δ ried out by Frik [2003], liquid sodium used as a work- sion, g0 is the gravity acceleration, T is the unit pertur- bation of the temperature T relative to the equilibrium ing medium can be considered as completely incom- η pressible. The dynamo mechanism operates in the profile T0, and is the magnetic diffusion coefficient. Earth’s outer core, consisting mostly of liquid iron and The Rossby number is defined as Ro = EPrÐ1. having a density changing, according to current ideas, Problem (3) becomes fully defined by introducing by 15Ð20% along the radius [Geomagnetism, 1988]. boundary conditions at r = ri, r0. Zero boundary condi- The situation with compressibility is similar in planets tions are used for the temperature perturbations T, and, of the solar system. Moreover, the coefficient of volume in conjunction with the profile T0 specified above, this expansion in the Earth’s core is too small (~2 × 10–5 KÐ1) corresponds to fixed values of the total temperature to generate heating-induced helicity in terms of T0 + T: (1, 0). Note that variation in spherically sym- Parker’s model. metric boundary conditions for T (e.g., fixed temperature or heat flux at boundaries or various forms of heat However, even the first 3-D models of the geody- source distribution in the core) affects weakly the gen- namo in the Boussinesq approximation had regimes erated magnetic field beginning from the onset of with nonzero large-scale helicity [Glatzmaier and Rob- developed convection [Sarson et al., 1997; Reshetnyak erts, 1995]. Below, we consider possible mechanisms and Steffen, 2005] but can dramatically change the of the helicity generation in such systems and compare regime of geomagnetic field reversals in the presence of our results with known models of the solar dynamo, in inhomogeneities of the heat flux at the core–mantle which the generation mechanisms of the α-effect have boundary ranging within a few tens of percent [Glatz- been discussed for many years [Krivodubskii, 1984; maier et al., 1999]. Therefore, without loss of general- Rüdiger and Kitchatinov, 1993]. ity in what follows, we accept the condition of the first

ri/r Ð 1 kind with fixed temperatures: T0 = ------at r0 = 1. EQUATIONS OF THE GEODYNAMO 1 Ð ri The impermeability condition Vr = 0 is accepted for Formulation of the Problem the velocity field. The tangential velocity components Vθ and Vϕ are subjected to the no-slip conditions Vθ = 0 We consider the geodynamo equations for an and Vϕ = 0 at the boundaries r r and r (the rigid core incompressible liquid (∇ á V = 0) in a spherical layer = i 0 ≤ ≤ in some models can rotate about the z axis) [Glatzmaier (ri r r0) rotating about the z axis at an angular veloc- Ω θ ϕ and Roberts, 1995], which is typical of rigid bound- ity in a spherical coordinate system (r, , ). We aries; an alternative to these boundary conditions for introduce the following units of measurement for the the tangential velocity components is the vanishing of V t P B κ L ↔ velocity , time , pressure , and magnetic field : / , τ τ τ 2 κ ρκ2 2 Ωρκµ tangential stresses at these boundaries: rθ, rϕ = 0 L / , /L , and 2 0, where L is the length unit, κ is the molecular heat conductivity coefficient, ρ is the [Kuang and Bloxham, 1997; Tilgner and Busse, 1997]. density, and µ is the magnetic constant; then, we have Boundary conditions of these types are used in models 0 of convection in stars with a free boundary [Chan- τ↔ ∂B Ð1 drasekhar, 1981] (exact expressions for in various ------= curl()VB× + q ∆B ∂t coordinate systems can be found in [Landau and Lif- shitz, 1975]). Such a type of boundary conditions, 1 Many geo- and astrophysical problems are characterized by con- somewhat strange for the Earth’s core, is related to the tinuous velocity field spectra and do not make the separation into following. It is known that, in the first case, the gener- large and small scales used in the dynamics of average fields. ated magnetic field concentrates near rigid boundaries

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 HYDRODYNAMIC HELICITY IN BOUSSINESQ-TYPE MODELS OF THE GEODYNAMO 451

Fig. 1. Results of calculations with E = 10Ð4, Pr = 1, and Ra = 102. The panels from left to right: the meridional cross sections of the nonaxisymmetric velocity components Vr (Ð4.0, 16.9), Vθ (Ð28.2, 28.4), and Vϕ (Ð33.6, 38.6) and the equatorial cross section of the nonaxisymmetric velocity component Vr (Ð23.9, 17.3). The numbers in parentheses bound the ranges of values of the components.

(see the comparison with a nonviscous model in the form of vertical rolls [Roberts, 1965; Busse, 1970; [Kuang and Bloxham, 1997]) and can be largely con- Boubnov and Golitsyn, 1995] in distinction to hexahe- trolled by processes in Ekman layers, in which intense dra observed in an incompressible liquid (see the shear flows exist and a nonzero helicity flux is present review in [Getling, 1999]). The rolls are known to at the rigid boundary. Given realistic values of the decrease in diameter (by the law ~E1/3) with increasing Ekman number E ~ 10–15, the thickness profile the rotation velocity of a sphere Ω. The realistic values of δ 1/2 –1 –15 Ekman layer is E = E L ~ 10 m, i.e., too small to the Ekman number in the Earth being E ~ 10 , the account for the existing large-scale (dipole) magnetic azimuthal wave number mcr at the convection genera- field of the Earth. The use of the turbulent value of the tion threshold (Ra = Racr) is estimated at ~105 [Jones, viscosity coefficient does not basically change the situ- 2000], which is beyond the possibilities of modern ation: in this case, ET ~ 10–9 and the layer density is numerical analysis and requires special models of tur- δT ~ 100 m, which is still at the resolution threshold of bulence [Reshetnyak and Steffen, 2004]. The situation E becomes even more complicated if real values of the modern computers. Another possibility is the use of cr 2 nonviscous conditions. In this case, the magnetic field Rayleigh number (Ra ~ 500 Ra [Jones, 2000]) are is generated in the main volume of the liquid core. In used. both cases, the magnetic field generated in the models Figure 1 presents characteristic cross sections of has features at the Earth’s surface similar to those velocity components demonstrating the formation of Geo- observed by archaeo- and paleomagnetologists [ vertical rolls at Ra = 100 and cr ~ 13. A further magnetism Ra , 1988]. Below, we apply in our calculations increase in the Rayleigh number not only decreases the the second type of boundary conditions in order to scale (Fig. 2) but also gives rise to a large-scale velocity remove boundary effects. Vacuum boundary conditions cr at the outer boundary r are used for the magnetic field component with wave numbers n < m well seen in the 0 field spectra (Fig. 3). Note that the widely accepted idea (their numerical finite-difference implementation is of an (albeit weak) attenuation of the poloidal magnetic described in [Hejda and Reshetnyak, 2000]). field at the surface of the liquid core ~e–0.1n [Lowes, To solve system (3) numerically, the method of control 1974] by no means implies (a) an attenuation of the volume [Patankar, 1980; Hejda and Reshetnyak, 2003, spectrum of the entire magnetic field (including the tor- 2004] was used and equations were solved in original × × oidal component) throughout the liquid core volume or physical variables on a 100 100 120 grid. PC clusters (b) an attenuation of the spectrum of kinetic energy (in and IBM SP 690+ supercomputers were used to the Earth, we have η ν 6 and the kinetic energy solve the problem (methods of using parallel proces- / = 10 spectrum is much more extended than the magnetic sors are described in detail in [Reshetnyak and Stef- energy spectrum). In the absence of superviscosity fen, 2005]). effects,3 a maximum in the kinetic energy spectrum is

Modeling Results 2 The Rayleigh number in the liquid core was estimated from the balance of the buoyancy and Coriolis forces. We consider the behavior of system (3) in the 3 In the model of Glatzmaier and Roberts [1995], the coefﬁcient of B ν ν absence of a magnetic ﬁeld ( = 0). An increase in the turbulent viscosity depends on the spectral number n as T = (1 + Rayleigh number Ra leads to convective instability in 0.075n3).

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Fig. 2. Results of calculations with E = 10–4, Pr = 1, and Ra = 5 × 102. The panels from left to right: the meridional cross sections of the velocity components Vr (Ð46.5, 104.6), Vθ (Ð111.1, 72.1), and Vϕ (Ð132.4, 136.8) and the equatorial cross section of the nonaxisymmetric velocity component Vr (Ð129.6, 128.0). The numbers in parentheses bound the ranges of values of the components. well observed in several dynamo models both with HELICITY GENERATION cr small Ra ~ 10 Ra [Kutzner and Cristensen, 2002; Sim- Helicity Models itev, 2004] and with developed convection with Ra Racr [Roberts and Glatzmaier, 2000]. Presently, several mechanisms of helicity generation are known. Before discussing the most widespread The broadening of the region of convection wave mechanisms, we should mention the possible genera- numbers is related to both linear effects (the number of tion of helicity due to reflection of Alfvén waves from modes increases at Ra > Racr) and the backward energy rigid boundaries [Moffat, 1978; Anufriev, 1991]. This transfer along the spectrum. The latter phenomenon is model once served as a basis for a special choice of the well consistent with the results of linear analysis (with- form of the α-effect in the geodynamo Z model [Bra- out nonlinear terms), according to which the Rayleigh ginsky and Roberts, 1987; Cupal and Hejda, 1992]. In numbers should be much greater than the generally terms of this model, the transition from the α-effect of cr α α θ accepted estimates Ra ~ 500 Ra in order to excite con- the form = 0cos that is classical in the star dynamo vection in the liquid core comparable in scale to the [Zeldovich et al., 1983] to the α-effect concentrated at core itself [Reshetnyak, 2005b]. the coreÐmantle boundary was instrumental in passing from the regime of traveling waves of the solar type to A characteristic feature of the two regimes consid- a stable dipole magnetic field such that, in the main vol- ered above is the separation of the liquid core into two ume of the liquid core, the field is directed along the regions (for details and literature, see [Jones, 2000]): rotation axis z, which gave the name to the Braginsky region I above (under) the rigid core bounded by the model. Taylor cylinder (TC) (the TC walls correspond to Stu- α artson layers) and region II outside the TC (Fig. 1). Note also the possible generation of the -effect by cr shear flows, for example, in surface Ekman layer of the Given the excitation threshold Ra = Ra and moderate atmosphere [Chkhetiani, 2001]. However, it is evident Rayleigh numbers (Fig. 1), convection develops in that the description of helicity and the α-effect in terms region II and is virtually absent in region I, but a rise in of other, volume models is preferable for the modeling the amplitude of heat sources (Fig. 2) shifts the convec- of large-scale magnetic fields. tion intensity maximum into region II. However, these regions are well resolved in both cases. On the whole, the ideas of developing volume models of helicity are based on the construction of a pseu- As noted above, helicity (1) is generated in models doscalar value obtained from the scalar product of the with an incompressible liquid (e.g., see [Glatzmaier ordinary (G) and axial (w) vectors. The Parker model and Roberts, 1995], and references to many publica- [Parker, 1979], described above and the most popular in tions can be found in [Simitev, 2004]). Distributions of astrophysics, relates G to a large-scale gradient of den- the hydrodynamic helicity χH averaged over time and sity and uses the vorticity w = curlv as the second vec- the azimuthal coordinate ϕ are presented in Fig. 5 for tor. However, this model cannot describe all possible the aforementioned two excitation regimes I and II. situations. Thus, in the presence of strong magnetic Below, we discuss the mechanisms of helicity genera- fields, one should take into account the back effect of a tion and compare the inferred results with data of other large-scale magnetic field, tending to suppress arising studies. small-scale flows. Then, in contrast to the hydrody-

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 HYDRODYNAMIC HELICITY IN BOUSSINESQ-TYPE MODELS OF THE GEODYNAMO 453 namic (in the given case, gyrotropic) helicity, there et al., 1980]. In a linear regime (small |V|), at 1Ω = 1z, arises the magnetic helicity χM, which decreases the ∂ V –1 Σ H M Σ H M the relation ------~ –Ro 1 × V yields total helicity χ = χ + χ , so that |χ | |χ | + |χ |. The ∂t z related suppression mechanism is described in [Vainsh- tein and Vainshtein, 1973; Krause and Rädler, 1980; ∂ Ð1⎛⎞∂E -----()V ⋅ curlV ∼ Ro ------k ÐV divV + [] curlVV× . Zeldovich et al., 1983], and the evolutionary equation ∂t ⎝⎠∂z z z for χM is presented in [Kleeorin et al., 2000]. Estimates (4) of the helicity suppression in the Earth’s liquid core are Historically, only the second term was initially taken given in [Reshetnyak and Sokolov, 2003]. into account, which is characteristic of gaseous bodies with high density gradients and a homogeneous distri- Although the Parker model was initially developed bution of kinetic energy. The continuity condition specially for the solar dynamo [Parker, 1979], subse- div(Vρ) = 0 yields divV = –V á G, where G = ∇logρ , quent observations of helioseismology (see the history and the helicity contribution of the second term of the problem in [Rüdiger and Kitchatinov, 1997]) pro- –1 amounts to –Ro (G á 1z). Evidently, with increasing vided additional constraints on the distribution of the → χH rotation velocity, Ro 0 and the helicity increases. helicity in the convective zone of the Sun: the helic- This mechanism describes well the generation of the χH ity should change its sign along the radius ( < 0 for α-effect in galaxies, where |G| > 1 [Zeldovich et al., large r and χH > 0 in the lower part of the convective 1983]. zone), which is obviously at variance with an ascending However, even in models of the solar dynamo and expanding eddy; this required a modification of [Krivodubskii, 1984; Rüdiger and Kitchatinov, 1993], numerical models [Yoshimura, 1975]. Moreover, the the change in the helicity sign in the lower part of the hypothesis on a compressible liquid in the lower part of convective zone could not be accounted for without the convective zone is inconsistent with modern ideas invoking the helicity generation due to the spatial inho- of the structure of the Sun. mogeneity of velocity field pulsations. In the general case, the expressions for the α-effect in the Sun have Another scenario of the generation of helicity and α ∇ ρ the α effect proposed in [Krivodubskii, 1984; Rüdiger the form ~ ln(v⊥ ), where v⊥ is the velocity pulsa- and Kitchatinov, 1993] was successfully applied in tion component perpendicular to the rotation axis.4 studies of the solar dynamo. According to this model, It is evident that the contribution of the first term can the α effect is determined by the kinetic energy gradient be fairly large in the planetary dynamo, where the mag- ∇ EK rather than the density gradient. Formally, this is a netic field is generated in a weakly compressible consequence of the fact that a pseudoscalar can be con- medium. Thus, the contribution of the first term in (4) ∇ Ω to the helicity generation can already be comparable structed from EK and the angular velocity vector . with that of the second term in the Earth’s core, whose This mechanism is interesting if only because spatial composition is accounted for by 95% molten iron and inhomogeneity of the kinetic energy (in both experi- whose density varies on the core scale by |G| ~ 0.2 [Bra- ments and astrophysical bodies) is observed much more ginsky and Roberts, 1995]. often than spatial inhomogeneity of density. It is worth Ω As regards the third term in (4), a more detailed noting that, in the case of an extremely fast rotation ( v v → ∞ α ≠ analysis of flow patterns shows that the product zcurlz ), the existence of 0 can be substantiated not- gives the largest contribution to the helicity and, withstanding the tendency of turbulence toward a 2-D accordingly, the third term can be neglected. pattern [Rüdiger, 1978].

These ideas of the generation of large-scale mag- Regimes of Small Ra netic ﬁelds by small-scale turbulence in an incompressible liquid do not contradict the well-known theoretical Convection at the excitation threshold and compar- geodynamo results [Glatzmaier and Roberts, 1995]. atively small Rayleigh numbers (Ra ~ 10 Racr) was Below, we demonstrate the formation of the α-effect in studied in detail by Busse (see the review in [Simitev, a rotating body with variable kinetic energy using, as an 2004]). The convection concentrates in vertical rolls in example, numerical models in the Boussinesq approxi- region II and is weak in region I. In this case, the so- mation. called weak dynamo is realized, when time-averaged magnetic and kinetic energies have the same order of magnitude. Sources of Helicity We consider the widely used ﬂow type independent in an Incompressible Liquid of z (with divv = 0) [Tilgner, 1997; Jones, 2000]. 4 Before analyzing more complex helicity models, we Since the value of turbulent pulsations in the lower convective H zone of the Sun is larger compared to the overlying region and the consider possible mechanisms of the χ generation by heat exchange is convective, rather than radiative, χH changes its the Coriolis force at a qualitative level [Vainshtein sign to the opposite.

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108 106 3 3

6 104 10 1 1 2 10 2 104 2 100

102 10–2

10–4 100 100 101 102 100 101 102 n n

ϕ Fig. 3. Fourier spectra of the -components of the velocity in the azimuthal direction for two regimes (Fig. 1): (1) Vr; (2) Vθ; (3) Vϕ.

∂ v Below, we note the stage at which the z dependence of NavierÐStokes equations: Ro curlz vú = ∂----- z + flows becomes necessary. A roll-like flow arising in the z ∆ northern hemisphere can be written by the formula E curlzv. The system is split at kz = 0, and the z component of the vorticity attenuates (the condition of col- 1 1 v ==()v ,,v v ( 2sin()πx cos() πy , linearity of the vectors z and W is inessential). The x y z (5) presence of finite gradients in z on the main scale L is Ð 2cos()πx sin() πy , Ð2sin()πx sin()) πy . necessary for the origination of rolls. The departure from geostrophism leads to nonvanishing gradients of The helicity of such a flow is always negative: kinetic energy, which corresponds to the first term in (4). The latter is a necessary (but not sufficient) condi- χH χH = Ð22ππsin2()x cos 2()πy tion for the generation of . The generation of large- (6) scale helicity is critically dependent on the relationship 2 2 2 2 Ð2 2ππcos ()x sin ()πy Ð 42ππsin ()x sin ()πy . between the phases of vz and wz, as is evident even from the fact that, making the replacement vz = –vz in (5), we According to properties of the equatorial symmetry of obtain χ → –χ. Given a layer infinite in z in the k-space, flows (Fig. 1), we have in the southern hemisphere (z < 0) we have at the excitation threshold for the neutral mode curlv(–z) = curlv(z) and vz(–z) = –vz(z) (for details, see ik c z | |2 [Busse, 1970]). This leads to a sign change at the equa- [Reshetnyak, 2005b]: h = v z* wz = ------vz . ()Ð1σ 2 tor: χH(–z) = –χH(z). This flow is geostrophic. However, E Pr + k the balance alone between the pressure and the Coriolis 2Prωk Then, we obtain χ = hc = ------z - , where ω = force is insufficient for actual realization of the Busse E()ω2 + k 4Pr2 rolls; the latter requires additionally the incorporation ∂ of the next terms in the expansion in powers of the –i -----v/v is the imaginary part of the growth rate. Since Ekman number [Busse, 1970], accounting for the buoy- ∂t ancy forces and the z dependence of fields. One of the the analysis of variance implies that two solutions with conditions following from the asymptotic analysis of ±k exist for a certain value of ω (waves traveling in such flows is the existence of finite gradients along the opposite directions), both positive (k > 0) and negative | | | | | | ≠ z z axis, kx ~ ky kz 0, where ki is a wave number (kz < 0) helicity can be generated. The total helicity van- (e.g., see [Reshetnyak, 2005b]), i.e., a departure from ishes, as is the case with the Alfvén waves in an infinite geostrophy. With reference to a problem in a sphere, layer [Moffat, 1978]. this means the influence of boundary conditions on the velocity field (Fig. 1). This statement can easily be The generation of nonzero helicity χH can be related understood, considering one of the three relations used to two effects [Busse, 1975, 1976]. These are the in the linear analysis of equations, namely, the relation Ekman layers and the inclination of the layer boundary, corresponding to the z component of the curl of the changing the height of rolls. In the first case, a nonzero

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 HYDRODYNAMIC HELICITY IN BOUSSINESQ-TYPE MODELS OF THE GEODYNAMO 455 normal component of the velocity ~E1/2 arises at the boundary of the Ekman layer, generating helicity (~E1/2). In the northern hemisphere, the helicity is positive at the outer boundary of the core and negative at the inner core boundary. The spatial scale of the helicity in the direction normal to the boundary is comparable δ 1/2 to the thickness of the Ekman layer E ~ E . This case is of no interest for the geodynamo theory. The second case is more interesting. It was shown that, using the decomposition in the height of rolls depending on the distance from the rotation axis, an increase or decrease in the roll height changes the helicity sign. The effect is associated with Rossby waves shifting the rolls in the radial direction, and this move- ment gives rise to their additional twisting (for more details, see [Busse, 2002]). This effect was usually considered in the outer part (II) of the core, where the roll height increases with decreasing distance from the s Fig. 4. Meridional cross section of the axisymmetric com- axis. We think that precisely this phenomenon is χH 2 responsible for the generation of the helicity χ inferred ponent of the helicity for the Rayleigh numbers Ra = 10 (–6.9 × 105, 6.9 × 105) in the left panel and Ra = 5 × 102 in [Kageyama and Sato, 1997] and from our calcula- × 7 × 7 tions at small Rayleigh numbers (Fig. 4). The effect of (–3.1 10 , 3.0 10 ) in the right panel. boundaries leads to a negative correlation between vz and wz. Unlike atmospheric cyclones and anticyclones Ω characterized by a positive correlation, the effect of boundaries yields an opposite helicity sign for region II. It is also worth noting that a decrease in the Prandtl number Pr → 0, shifting rolls into the equatorial zone (large s) [Zhang, 1993], can lead to a similar shift of χH into the equator area. However, this effect is of no sig- I niﬁcance for the Earth’s core because, in models of both thermal convection (Pr ~ 0.1) and concentration convection (Pr ~ 10), the resulting displacement is small, particularly, at large Reynolds numbers Re (Re = VL 9 II II ------ν- ~ 10 in the Earth). The values of the turbulent Prandtl number PrT are ~O(1) in both cases.

Moffat Estimates I Before analyzing the regime of vigorous convection, we consider some interesting consequences of the Moffat estimates with reference to the small-Ra convection model discussed above. Throughout this sec- z tion, we assume that the distribution of the Busse rolls is homogeneous in the plane (x, y), perpendicular to the rotation axis z, and the fields do not vary along this Fig. 5. The sense of the meridional circulation is shown in 5 the upper part of the figure and the angular velocity of the axis. liquid rotation Vϕ/s is shown in the lower part. We decompose the velocity field into the longitudi- nal component along the z axis and a transverse component in the plane (x, y): Then, due to the homogeneity of the roll distribution in the plane (x, y), the expression for the mean helicity VV==|| + V⊥, V|| 1zV z. (7) takes the form [Moffat, 1978] H 5 The statistical homogeneity of a flow in the plane (x, y) corre- ==〈〉V ⋅ curlV 〈〉V⊥ ⋅ curlV⊥ sponds to rather large values of Ra, so that rolls fill a spatial (8) region much greater in size than their diameter. +2〈〉V|| ⋅ curlV⊥ .

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 456 RESHETNYAK ≡ ≡ The second term (Vzwz V||curlV⊥) is already consid- layer (1z 1W), the application of the curl operation to ered above. The first term in (8) can be estimated from (9) and the scalar multiplication by V yield the balance of the buoyancy Coriolis forces in (3): ∂ 1 ×∇V = ÐRap + T1 . (9) ----- E = Ra∇T ⋅ ()1 ⋅ V . (11) z z ∂z k z After applying the curl operation to (9), multiplying the result by 1z × V, and taking into consideration (7) (for Assuming the axial symmetry and writing out the right- details, see [Moffat, 1978]), we obtain an estimate for hand side of (11) in the spherical system of coordinates ϕ ∇ ∇ ∇ the first term in (8): (s, , z), we obtain ( sT, 0, zT) á (–Vϕ, Vs, 0) = –Vϕ sT, ∇ T ∂ where s is the so-called Archimedes wind. Estimate V z 〈〉V⊥ ⋅ curlV⊥ = ÐRa T------. (10) (11) does not contradict the numerical results reported ∂z in [Glatzmaier and Roberts, 1995] because the temper- ∇ As before, we see that the dependence on z is necessary ature in region I rises toward the axis, sT < 0, and the for the helicity generation. Note also that the limiting Coriolis force hinders to a lesser extent the motion and transition from the periodic flow considered above to a the heat transfer from the hotter rigid core toward the ∂ random flow and Moffat estimates (10) can be made outer core boundary and the sign change in ----- E is only accurate to coefficients of the order of unity (Zel- ∂z k dovich [1984] describes in more detail a similar case of determined by the inversion of the azimuthal velocity estimating the coefficients of turbulent diffusion for Vϕ(z) (see Fig. 5; in the case of a fast rotation, the azi- random and periodic flows). muthal velocity component usually has a maximum energy). As noted above, a kinetic energy varying along Regimes of Large Ra the z coordinate is necessary for the generation of helicity. The following scenario is possible: if region I is hot- At large Ra, the roll mechanism of heat transfer in ter than region II, the average of the component of Vz is region II becomes insufficient and convection moves positive (in the northern hemisphere). This corresponds into region I, where flows driven by the buoyancy to the flow of the liquid into the lower part of the TC forces are parallel to the rotation axis and the Coriolis (and away from its upper part) through the Ekman layer force hinders to a lesser extent the motion and the heat and the TC walls. The latter implies the introduction of transport from the hotter rigid core to the outer bound- a positive (negative) torque and, as a result, a positive ary. Overall, the mean temperature in region I is higher (negative) torsion in the lower (upper) part of the TC, as (T > 0) than in region II; however, along with the mean is observed in Fig. 5. However, two problems arise in ascending flow, the roll structure is also present in this this case. These are the questions of why a positive region. Accurate estimates of the axisymmetric and helicity is not observed near the equator at small Ra and asymmetric components can be found in [Glatzmaier why, for the Ekman numbers considered here, the helic- and Roberts, 1995], but their ratio can depend strongly ity has a scale much smaller than the observed scale. on the superviscosity model in use [Zhang and Jones, 1997]. To narrow the range of possible situations, we Modeling at large Ra ~ 103 Racr being complicated address thermal convection with large Rayleigh num- by the development of numerical instabilities, earlier bers (Ra = 500) and nonviscous boundary conditions. studies of the 3-D dynamo included artificial suppres- As noted above, an increase in Ra shifts convection into sion of high frequencies with the use of a superviscosity the TC (Figs. 1, 2). Although the use of nonviscous [Glatzmaier and Roberts, 1995; Kuang and Bloxham, boundary conditions resulted in a sign-constant angular 1997]. In the case of a vanishing relative velocity at the velocity of the TC liquid, which was observed in boundary of the liquid core, a change in the helicity [Glatzmaier and Roberts, 1995] but was not observed in [Kuang and Bloxham, 1997], the helicity distribution sign in region I is observed with increasing r: in the northern hemisphere, the helicity sign is positive in the remained qualitatively the same (Fig. 4): the TC value χH lower part and negative in the upper part. This is the of changes its sign in the vertical direction. The basic distinction from the small-Ra case considered numerical experiments discussed above demonstrate χH that the elimination of the axisymmetric component of above, in which the helicity does not change its sign ϕ |χH| the velocity V changes insignificantly (by ~15%) the - throughout the hemisphere. Maximums of are close χH to rigid boundaries. It is also known that, in the model averaged helicity . The termwise analysis of the χH of Glatzmaier and Roberts [1995], the axisymmetric expression for component accounts for 80% of the total velocity field ∂ ∂ and the angular velocity changes its sign in the TC: χH V r ⎛⎞()θ V θ = ------⎝⎠------V ϕ sin Ð ------Vϕ(z)/s > 0 (s = rsinθ) in the lower part and <0 in the rsinθ ∂θ ∂ϕ upper part (Fig. 5). (12) V ∂V ∂ V ∂ ∂ θ⎛⎞1 r () ϕ⎛⎞()V r Now, we compare results of these calculations with + ------⎝⎠------Ð ----- rVϕ + ------⎝⎠----- rVθ Ð ------the vertical variation in the kinetic energy. In a plane r sinθ ∂ϕ ∂r r ∂r ∂θ

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 HYDRODYNAMIC HELICITY IN BOUSSINESQ-TYPE MODELS OF THE GEODYNAMO 457 in this model shows that the first of the three terms in dient alone is insufficient for the realization of large- (12) makes the greatest contribution to the formation of scale helicity. This condition is necessary for the forma- two χH structures of opposite signs in the TC; i.e., as in tion of the roll flow structure, but the correlation of vor- the cases of small Ra considered above, we deal with a ticity with the vertical velocity component requires nonzero correlation of small-scale cyclonic flows additional conditions, in particular, an inclination of the resulting in a large-scale effect but with a smaller scale volume boundaries. χH along the z axis: changes its sign. As seen from Fig. 2, Above, we did not touch on the problem of calculat- eddies in the lower and upper regions of the northern ing the α-effect (2). Practical use of the above helicity hemisphere rotate in opposite senses, but the direction estimates in large-scale dynamo models (such as the of the vertical velocity component remains the same all dynamo of mean fields [Krause and Rädler, 1980]) along one roll. requires information on the spectra of the fields V and B. Since the calculations were performed for nonvis- Using the simplest estimate for the mixing length, we cous boundary conditions, we think that the positive have τH = l/v, i.e., α ~ –v/3. Assuming the spectrum of helicity arises at the boundary of the rigid core due to the velocity field to be a white noise, vn = const = Vwd, the variation in the roll height associated with the we obtain that the magnetic Reynolds number rm = boundary curvature [Busse, 1975]: with an increase in v/(nη) on a scale of 1/n is ~1 even for a scale of 10–3L, the distance from the z axis caused by the Rossby so that the turbulence-driven generation of the mag- waves, the roll height increases, which corresponds to netic field is ineffective. The latter imposes a constraint the positive sign of the helicity near the rigid core. The on the maximum number of rolls: n 3 [Jones, spatial scale of the distributions of positive and negative max ~ 10 helicities is ~(r – r )/2 (Fig. 4). 2000]. In the case of turbulence involving a cascade of 0 i rolls of different diameters, this constraint agrees with the estimate of the energy-carrying scale for the term DISCUSSION curl(αB) in the induction equation. The resulting ampli- α tude of the -effect (~Vwd/3) and the related value of the Apparently, direct numerical modeling fails to pro- α –15 –1/3 × 7 L vide regimes with E ~ 10 and Ra ~ 500E = 5 10 dynamo number Rα = ------used in the αω-dynamo and one can only hope that the model results already η belong to the asymptotic regime E 0, Ra/Racr = models are comparable to estimates derived in [Anu- const. However, the fact that the effect of the helicity friev et al., 1997]. The use of models with a decreasing –γ formation described above depends only on the tilt of spectrum (vn ~ n ), i.e., the Kolmogorov turbulence the boundary and is independent of E, makes the given with γ = 1/3, increases the energy-carrying scale mechanism attractive for modeling the helicity at real- (decreases nmax). istic values of parameters. The helicity generation mechanisms considered above can exist in the Earth’s liquid core along with the ACKNOWLEDGMENTS traditional mechanism proposed by Parker [1979] (see I am grateful to F.H. Busse and D.D. Sokolov for also [Shimizu and Loper, 2000]). The concurrent oper- help in my work. This work was supported by the Rus- ation of both mechanisms leads to an increase in the sian Foundation for Basic Research, project no. N03- helicity near the outer boundary and they weaken each 05-64074, and INTAS, grant no. N03-51-5807. other near the rigid core; i.e., they operate in an antiphase mode. Since the helicity in both mechanisms is antisymmetric with respect to the equatorial plane, REFERENCES the total helicity also possesses this property. α It is also worth noting that the form of the α-effect, 1. A. P. Anufriev, “An -Effect on the Core Mantle Bound- ary,” Geophys. Astrophys. Fluid Dynamics 57, 135Ð146 concentrated near the coreÐmantle boundary in the Bra- (1991). ginsky Z model and observed in our study in the TC, can be interpreted in terms of not only the reflection of 2. A. P. Anufriev, M. Yu. Reshetnyak, and D. D. Sokolov, Alfvén waves from a rigid boundary but also purely “Estimation of the Dynamo Number in the Turbulent α- hydrodynamic processes near rigid boundaries. How- Effect for the Earth’s Liquid Core,” Geomagn. Aeron. ever, in this case, the model should be complemented 37, 141Ð146 (1997). by helicity of an opposite sign near the boundary of the 3. J. M. Aurnou, D. Brito, and P. L. Olson, “Mechanics of rigid core (note that initially the Z model was devoid of Inner Core Super-Rotation,” Phys. Earth Planet. Inter. a rigid core). The ideas of sources of α- and Ω-effects 23, 3401Ð3407 (1996). concentrated in different spatial regions were also 4. B. M. Boubnov and G. S. Golitsyn, Convection in Rotat- applied in studies of the solar dynamo [Stix, 1971]. ing Fluids (Kluwer, London, 1995). As regards the formal similarity between the 5. S. I. Braginsky and P. H. Roberts, “Model Z Geody- inferred dependences and solar dynamo models, we namo,” Geophys. Astrophys. Fluid Dynamics 38, 327Ð should note that the existence of the kinetic energy gra- 349 (1987).

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 458 RESHETNYAK

6. S. I. Braginsky and P. H. Roberts, “Equations Governing Dynamo Generated Galactic Magnetic Field,” Astron. Convection in Earth’s Core and the Geodynamo,” Geo- Astrophys. 361, L5ÐL8 (2000). phys. Astrophys. Fluid Dynamics 79, 1Ð97 (1995). 27. F. Krause and K.-H. Rädler, Mean Field Magnetohydro- 7. E. C. Bullard and H. Gellman, “Homogeneous Dynamos dynamics and Dynamo Theory (Akademie-Verlag, Ber- and Terrestrial Magnetism,” Phil. Trans. R. Soc. London lin, 1980). A247, 213Ð278 (1954). 28. V. I. Krivodubskii, “Source Intensity of Magnetic Fields 8. F. H. Busse, “Thermal Instabilities in Rapidly Rotating of the Solar αω-Dynamo,” Astron. Zh. 61 (3), 540Ð548 Systems,” J. Fluid Mech. 44, 441Ð460 (1970). (1984). 9. F. H. Busse, “A Model of the Geodynamo,” Geophys. J. 29. W. Kuang and J. Bloxham, “An Earth-Like Numerical R. Astron. Soc. 42, 437Ð459 (1975). Dynamo Model,” Nature 389, 371Ð374 (1997). 10. F. H. Busse, “A Model of the Geodynamo,” Phys. Earth 30. W. Kuang and J. Bloxham, “Numerical Modeling of Planet. Inter. 12 (4), 350Ð358 (1976). Magnetohydrodynamic Convection in a Rapidly Rotat- 11. F. H. Busse, “Convective Flows in Rapidly Rotating ing Spherical Shell: Weak and Strong Field Dynamo Spheres and Their Dynamo Action,” Phys. Fluids 14 (4), Action,” J. Comput. Phys. 153, 51Ð81 (1999). 1301Ð1314 (2002). 31. M. V. Kurganskii, Introduction to Large Scale Dynamics 12. S. Chandrasekhar, Hydrodynamics and Hydromagnetic of the Atmosphere: Adiabatic Invariants and Their Stability (Dover, New York, 1981). Applications (Gidrometeoizdat, St. Petersburg, 1993) 13. O. G. Chkhetiani, “On the Helical Structure of the [in Russian]. Ekman Boundary Layer,” Izv. Akad. Nauk, Fiz. Atmosf. 32. C. Kutzner and U. R. Cristensen, “From Stable Dipolar Okeana 37 (5), 614Ð620 (2001). towards Reversing Numerical Dynamos,” Phys. Earth 14. I. Cupal and P. Hejda, “Magnetic Field and α-Effect in Planet. Inter. 131, 29Ð45 (2002). Model Z,” Geophys. Astrophys. Fluid Dynamics 67, 87Ð 33. L. D. Landau and E. M. Lifshitz, Course of Theoretical 97 (1992). Physics, Vol. 2: The Classical Theory of Fields (Nauka, 15. F. V. Dolzhanskii, “On Mechanical Originals of Funda- Moscow, 1988; Pergamon, Oxford, 1975). mental Hydrodynamic Invariants,” Izv. Akad. Nauk, Fiz. 34. M. Lesieur, Turbulence in Fluids (Kluwer, London, Atmosf. Okeana 37 (4), 446Ð458 (2001). 1997). 16. P. G. Frik, Turbulence: Approaches and Models (Inst. 35. F. J. Lowes, “Spatial Power Spectrum of the Main Geo- Komp’yuternykh Issledovanii, Izhevsk, 2003) [in Rus- magnetic Field,” Geophys. J. R. Astron. Soc. 36, 717Ð sian]. 725 (1974). 17. Geomagnetism, Ed. by J. A. Jacobs (Academic, New 36. H. K. Moffat, Magnetic Field Generation in Electrically York, 1988). Conducting Fluids (Cambridge Univ., Cambridge, 18. A. V. Getling, RayleighÐBenard Convection: Structures 1978). and Dynamics (Editorial URSS, Moscow, 1999) [in Rus- 37. E. N. Parker, Cosmical Magnetic Fields: Their Origin sian]. and Their Activity (Clarendon Press, Oxford, 1979; Mir, 19. G. A. Glatzmaier and P. H. Roberts, “A Three-Dimen- Moscow, 1982). sional Convective Dynamo Solution with Rotating and 38. S. V. Patankar, Numerical Heat Transfer a Fluid Flow Finitely Conducting Inner Core and Mantle,” Phys. (Taylor & Francis, New York, 1980). Earth Planet. Inter. 91, 63Ð75 (1995). 39. M. Yu. Reshetnyak, “Estimation of the Turbulent Viscos- 20. G. A. Glatzmaier, R. S. Coe, L. Hongre, and P. Roberts, ity in the Liquid Core of the Earth,” Dokl. Akad. Nauk “The Role of the Earth’s Mantle in Controlling the Fre- 400 (1), 105Ð109 (2005a). quency of Geomagnetic Reversals,” Nature 401, 885Ð 40. M. Yu. Reshetnyak, “Dynamo Catastrophe, or Why the 890 (1999). Geomagnetic Field is So Long-Lived,” Geomagn. 21. P. Hejda and M. Reshetnyak, “The Grid-Spectral Aeron. 45 (4), 571Ð575 (2005b). Approach to 3-D Geodynamo Modeling,” Computers & 41. M. Yu. Reshetnyak and D. D. Sokolov, “Geomagnetic Geosciences 26, 167Ð175 (2000). Field Intensity and Suppression of Helicity in the Geo- 22. P. Hejda and M. Reshetnyak, “Control Volume Method dynamo,” Fiz. Zemli, No. 9, 82–86 (2003) [Izvestiya, for the Dynamo Problem in the Sphere with the Free Phys. Solid Earth 39, 774Ð777 (2003). Rotating Inner Core,” Studia Geophys. Geodet. 47, 147Ð 42. M. Reshetnyak and B. Steffen, “The Subgrid Problem of 159 (2003). Thermal Convection in the Earth’s Liquid Core,” 23. P. Hejda and M. Reshetnyak, “Control Volume Method Numerical Methods and Programming 5, 41Ð45 (2004). for the Thermal Convection Problem in a Rotating http://www.srcc.msu.su/num-meth/english/index.html. Spherical Shell: Test on the Benchmark Solution,” Stu- 43. M. Reshetnyak and B. Steffen, “Dynamo Model in the dia Geophys. Geodet. 48, 741Ð746 (2004). Spherical Shell,” Numerical Methods and Programming 24. C. A. Jones, “Convection-Driven Geodynamo Models,” 6, 27Ð32 (2005). http://www.srcc.msu.su/num- Phil. Trans. R. Soc. London A358, 873Ð897 (2000). meth/english/index.html. 25. A. Kageyama and T. Sato, “Velocity and Magnetic Field 44. P. H. Roberts, “On the Thermal Instability of a Highly Structures in a Magnetohydrodynamic Dynamo,” Phys. Rotating Fluid Sphere,” Astrophys. J. 141, 240Ð250 Plasmas 4 (5), 1569Ð1575 (1997). (1965). 26. N. Kleeorin, D. Moss, I. Rogachevskii, and D. Sokoloff, 45. P. H. Roberts and G. A. Glatzmaier, “A Test of the Fro- “Helicity Balance and Steady-State Strength of the zen-Flux Approximation Using a New Geodynamo

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006 HYDRODYNAMIC HELICITY IN BOUSSINESQ-TYPE MODELS OF THE GEODYNAMO 459

Model,” Phil. Trans. R. Soc. London 358, 1109Ð1121 54. A. Tilgner and F. H. Busse, “Finite Amplitude in Rotat- (2000). ing Spherical Fluid Shells,” J. Fluid Mech. 332, 359Ð376 46. R. Rüdiger, “On the α-Effect for Slow and Fast Rota- (1997). tion,” Astron. Nachr. 299 (4), 217Ð222 (1978). 55. S. I. Vainshtein, Magnetic Fields in Cosmos (Nauka, Moscow, 1983) [in Russian]. 47. R. Rüdiger and L. L. Kitchatinov, “α-Effect and α- 56. L. L. Vainshtein and S. I. Vainshtein, “On the Possibility Quenching,” Astron. Astrophys. 269, 581Ð588 (1993). of a Linear Turbulent Dynamo,” Geomagn. Aeron. 13 48. R. Rüdiger and L. L. Kitchatinov, “The Slender Solar (1), 149Ð153 (1973). Tachocline: A Magnetic Model,” Astron. Nachr. 318, 57. S. I. Vainshtein, Ya. B. Zeldovich, and A. A. Ruzmaikin, 273Ð279 (1997). Turbulent Dynamo in Astrophysics (Nauka, Moscow, 49. G. R. Sarson, C. A. Jones, and A. W. Longbottom, “The 1980) [in Russian]. Inﬂuence of the Boundary Region on the Geodynamo,” 58. H. Yoshimura, “A Model of the Solar Cycle Driven by Phys. Earth Planet. Inter. 101, 13Ð32 (1997). the Dynamo Action of the Global Convection in the 50. H. Shimizu and D. E. Loper, “Small-Scale Helicity and Solar Convection Zone,” Astrophys. J. Suppl. 29, 467Ð α-Effect in the Earth’s Core,” Phys. Earth Planet. Inter. 494 (1975). 121, 139Ð155 (2000). 59. Ya. B. Zeldovich, Selected Works: Chemical Physics and Fluid Dynamics (Nauka, Moscow, 1984) [in Russian]. 51. R. Simitev, Convection and Magnetic Field Generation 60. Ya. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokoloff, in Rotating Spherical Fluid Shells, Ph. D., Bayreuth: Magnetic Fields in Astrophysics (Gordon and Breach, Univ. Bayreuth, 2004, p. 193. http://www.phy.uni- New York, 1983). bayreuth.de/theo/tp4/members/simitev.html. 61. K. Zhang, “On Equatorially Trapped Boundary Inertial 52. M. Stix, “A Nonaxisymmetric α-Effect Dynamo,” Waves,” J. Fluid Mech. 248, 203Ð217 (1993). Astron. Astrophys. 13 203Ð208 (1971). 62. K. Zhang and C. A. Jones, “The Effect of Hyperviscosity 53. A. Tilgner, “A Kinematic Dynamo with a Small Scale on Geodynamo Models,” Geophys. Res. Lett. 24, 2869Ð Velocity Field,” Phys. Lett. A226, 75Ð79 (1997). 2872 (1997).

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