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RUSSIAN JOURNAL OF EARTH SCIENCES, VOL. 14, ES2001, doi:10.2205/2014ES000539, 2014
The mean-field dynamo model in geodynamo
M. Yu. Reshetnyak1
Received 28 July 2014; accepted 30 July 2014; published 8 August 2014.
The 2D Parker's mean-field dynamo equations with a various distributions of the a- and w-effects are considered. We show that smooth profiles of a and w can produce dipole configuration of the magnetic field with the realistic magnetic energy spectrum. We emphasize that fluctuating a-effect leads to increase of the magnetic energy at the small scales, breaking the dipole configuration of the field. The considered geostrophic profiles of a and w correspond to the small-scale polarwards/equatorwards travelling waves with the small dipole field contribution. The same result is observed for the dynamic form of the a-quenching, where two branches of the weak and strong solution coexist. KEYWORDS: Earth's liquid core; turbulence; geodynamo.
Citation: Reshetnyak, M. Yu. (2014), The mean-field dynamo model in geodynamo, Russ. J. Earth. Sci., 14, ES2001, doi:10.2205/2014ES000539.
1. Introduction
The last decades demonstrated impressive success of the three-dimensional geodynamo modeling. It appears that combination of the compositional and thermal convection can drive the dynamo mechanism, transforming the heat and kinetic energies to the energy of the magnetic field. To the moment, the various 3D geodynamo models can reproduce the main features of the observable geomagnetic field: the dipole structure, reversals and excursions of the field, as well as the regimes without reversals, which correspond to the chrons, well known in palaeomagnetism [Roberts and King, 2013].
However analysis of the data simulated in the 3D models sometimes is not easier task rather the analysis of the observations itself. Moreover, due to specific of the 3D modeling, which requires the detailed resolution of the small-scaled turbulence, it appears impossible to reproduce the long-time sequences of the magnetic field evolution, compared with the paleo- and archemagnetic observations. Note that from the point of view of the observer, information that can be derived from the bulk of 3D data in the models, is excessive, because it can not be verified by the observations with the pure resolution.
The tendency of the geodynamo development only proves this statement: to reach the desired parameter regimes of the magnetohydrodynamic (MHD) process in the turbulent liq-
1 Schmidt Institute of Physics of the Earth of the Russian
Academy of Sciences, Moscow, Russia
Copyright 2014 by the Geophysical Center RAS.
http://elpub.wdcb.ru/journals/rjes/doi/2014ES000539.html
uid core one needs at least 1024 grid points for the data simulations that corresponds to the Reynolds number Re = 109. In the same moment, the direct observations of the geomagnetic field, limited by the screening of the low-conductive mantle, are bounded with the first decade of the spherical functions (more precisely n < 13) [Finlay et al., 2010]. It means that scale to scale comparison of 3D simulations with observations is only possible in the negligible part of the spatial scales, involved into the simulations. On the other hand, the time scales in 3D models are order of magnitude shorter than the geological times.
This discrepancy results in renovation of the quite old mean field approach in geodynamo, which is able to reproduce behavior of the large-scale magnetic field. In its turn, simulated large-scale magnetic field already can be easily compared with the observations. Due to reduction of the 3D basic equations to the axi-symmetric form, this approach permits to simulate long-term evolution of the magnetic field, compared with the palaeomagnetic records.
The mean-field theory was developed by the two independent scientific groups. The exhaustive theoretical background was elaborated in the German group [Krause and Radler, 1980], mostly concentrated on the astrophysical applications. The main result of the theory is the description of the large-scale magnetic field generation with the conductive turbulent medium and velocity shear. The back-reaction (or quenching) of the magnetic field onto the flow was introduced by the damping of the turbulence.
The other, geophysical approach, developed by the Russian scientist S. I. Braginsky, included influence of the magnetic field onto the large-scale velocity field. The famous geodynamo Z-model could reproduce the dipole structure of the magnetic field and made a remarkable insight into the physics of the liquid core [Braginsky, 1975]. One of the cru-
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cial points of this model is existence of the strong magnetic wind, which corresponds to the large magnetic field counterpart to the azimuthal force in the Navier-Stokes equation. That was the reason of the too strong toroidal magnetic field, compared to the poloidal part.
Only latter it was recognised that influence of the magnetic field on the flow is a very delicate process [Brandenburg and Subramanian, 2005; Hejda and Reshetnyak, 2010]: magnetic field does not change the cyclonic form of the flow essentially, as well as it does not produce too large azimuthal velocity [Jones, 2000]. This is the motivation to consider the classical mean-field dynamo equations without back-reaction of the magnetic field on the large-scale flow, using only a-quenching, concerned with the damping of the turbulence by the large-scale magnetic field. So far the kinetic energy of the turbulence is smaller than that of the large-scale velocity, suppression of the turbulence looks more acceptable.
This approach is supported by the new knowledge on the hydrodynamic of the liquid core: the spatial distribution of the differential rotation and kinetic helicity in the rotating spherical shell, where the geostrophic state holds [Reshet-nyak, 2010]. The other point is the study of the more complex quenching mechanisms of the a-effect [Kleeorin et al., 1995], developed after the first success of the mean-field theory, and its influence on the magnetic dipole behavior.
We also consider applications of the popular approach of the fluctuating a-effect [Hoyng, 1993], and discuss constraints on the amplitude of such fluctuations, which follows from the form of the spatial spectrum of the geomagnetic field [Langel, 1987].
2. Basic Equations and Methods of Solution
The mean magnetic field B is governed by the induction equation
8 B
~dt
V x (a B + V x B - ri rotB) ,
(1)
where V is the large-scale velocity field, a is the a-effect, and q is a magnetic diffusion.
The magnetic field B = (Bp, Bk) has two parts: the poloidal component Bp = V x A, and the toroidal component Bk, where A is the vector potential of the magnetic field.
The principal point of the mean-field dynamo theory is the separation of the physical fields onto the large- and small-scale counterparts. Information on the large-scale velocity field is described by V, and on the small-scale fields fluctuations by the a-effect.
Usually, it is supposed that the mean field B has axial symmetry. This assumption follows from the effect of the differential rotation, which suppresses deviations of the frozen magnetic field into the flow from the axial symmetry.
Due to the axial symmetry of the magnetic field, vector potential A and Bk have the only one azimuthal component in the spherical system of coordinates (r, 6, p): A(r, 6) =
(0, 0, A), and Bt(r, 6) = (0, 0, B). Then the poloidal field can be written in the form:
Bp
1
8
r sinö 88
(A siné»)
18, . — — (rA), r or
(2)
In terms of scalars A and B Eq (1) is reduced to the following system of equations:
8AA =aB + (V x B)^ + ( V2 - 2 . at ^ \ r2 sin2 0,
8B / 1
— = rotv (a B + V x B)+ V2 - 2 ; 2 /j ot ^ \ r2 Sin2 0 ,
(V - 1 ^ A
2 sin2
fv2 - -JL_)
2 sin2
(3)
B,
where the subscript p corresponds to the azimuthal component of the vector.
Eqs (3), solved in the spherical shell r¿ < r < r0 with r¿ = 0.35, r 0 = 1, are closed with the pseudo-vacuum boundary d
conditions: B = 0, and — (rA) = 0 at r¿ and r0 and A = or
B = 0 at the axis of rotation 6 = 0, n. The simplified form of the vacuum boundary condition for A is well adopted in dynamo community, and presents a good approximation of the boundary with the non-conductive medium [Jouve et al., 2008]. The reason why the vacuum boundary condition is used at the inner core boundary is discussed in [Reshetnyak, 2013] and concerned with the weak influence of the inner core on the reversals statistics of the magnetic field [Wicht, 2002].
In the general case velocity V is a three-dimensional vector, as a function of r and 6. Further we consider only the effect of the differential rotation, concerned with the p component of V, leaving the input of the meridional circulation ( Vr, Vg) out of the scope of the paper.
For the quite large amplitudes of a and V solution (A, B) grows exponentially, and one needs to introduce the feedback of the magnetic field onto the sources of the input energy a and V. As we already mentioned above, we concentrate our study on the feedback of the magnetic field onto the a-effect, responsible on the production of the large-scale poloidal magnetic fields by the small-scaled turbulence. This approach let us to bypass solution of the Navier-Stokes equation, which, at least in the geodynamo, is the most difficult part of the full dynamo problem. We recall that turbulent convection presents at the small scales, where the magnetic field is already absent due to the high magnetic diffusion. The ratio of the diffusion scales of the velocity and magnetic fields is of the order of the Roberts number q = 10-5, which is quite small in the liquid core. However the magnetic field is not generated at the small-scaled part of the kinetic energy spectrum, to get a self-consistent solution for the velocity field, one needs to solve the Navier-Stokes equation in the full range of scales. This task is still out of reach of the modern computer facilities.
Here we specify two forms of the a-quenching. The first one, the so-called algebraic quenching, originates from the simple idea of the damping of the a-effect's amplitude with the mean magnetic field:
(
1 +
Em(r
(4)
E °
a =
where Em = B2/2 is the magnetic energy, and E°n is the constant parameter. The choice of this parameter relates to our assumptions on the ratio of the kinetic to magnetic energies, and depends strongly on the angular rotation of the body [Reshetnyak and Sokoloff, 2003].
The more sophisticated form of the a-quenching follows from [Pouquet at al., 1976], [Zeldovich at al., 1990], where influence of the magnetic field onto the a-effect was described by the magnetic pat of the a-effect, so that the total effect is the sum: a = ah + am. Here ah and am are the hydrodynamic (the so-called kinetic a-effect) and magnetic parts, correspondingly. The damping of a means generation of am with the opposite sign to ah. This idea was formulated latter in the form of the evolutionary equation for am [Kleeorin et al., 1995], see for details Section 6. This kind of the a-quenching, derived from the basic MHD equations, leads to oscillations in the system, and was used in the solar dynamo to mimic the solar cycle of the magnetic activity.
The differential operators in Eqs. (3)-(4) were approximated with the second-order central-differences scheme in space, and integrated in time using the second-order Runge-Kutta method. These algorithms resulted in C++ object oriented code with use of Blitz++ C++ library for the easier compact operations with the arrays. The post-processor graphic visualization was organized using the Python graphic library MatPlotlib. All simulations were done under the Linux OS.
The code passed the set of the benchmarks. The first one is the free-decay mode test for the diffusion operator in the equation:
dA ~dt
(V -
V r2 sin2 0 J
A,
(5)
d
A = J* (ji(VXr) + Cyi(VXr)) Pi
(6)
The other test was the benchmark on the threshold of the magnetic field generation, the Case A' from [Jouve et al., 2008], which was also passed successfully.
3. Simple Forms of a-u Profiles
We start from the simple forms of the a-effect and az-imuthal velocity Vv, adopted in the mean-field dynamo.
From the general arguments it is known that the a-effect has the dipole symmetry in respect to the equator plane. We also assume that it is positive in the northern hemisphere, so that
—— ^ sin(20), (7)
ra - ri )
= Ca sin
in ^
where Ca is a positive constant. This assumption is in agreement with that fact that kinetic helicity \ is negative in the northern hemisphere and a ~ — \ [Krause and Radler, 1980]. In (7) a vanishes at the poles, which, as we see below, is also the good approximation of the real a-effect, derived from 3D models [Reshetnyak, 2010]).
For the azimuthal velocity we take
Vv = C^ (r - ri)(ra - r)
-0.7-1 (* - I) 2 •
; V I / si
D sin(0), (8)
with A = 0 at the axis and at r = ri, and — (r A) = 0 at
or
ra.
Simple analytic solution of (5) for testing can be written in the form:
where Pf is the associated Legendre polynomial of degree 1, and order 1, and ji, yi are the spherical Bessel functions of the first and second kind. Note, that in contrast to the scalar Laplace equation, where the axially symmetric meridional part of the solution is described by Pj°, the order of our vector diffusion operator's eigenfunctions is shifted by one, and corresponds to Pf.
Putting expression for A (6) in (5), and using boundary conditions for A at the radial boundaries, one has condition of solvability for A. Solution of this transcendent equation, using package of the analytic algebra SymPy for the Python, gives A = 4.8732823108648490873, that leads to C = 0.380157168844938 and 7 = -23.74888048138824458988. This estimate of A, C, and 7 is enough to satisfy to the boundary conditions with the double precision accuracy, used in the program. Using (6) as the initial condition, we simulated Eq (5) and obtained the decay rate equal to the analytic 7 with the accuracy up to 0.5% for Nr x Ng mesh grid points, with Nr = Ng = 101.
where C^ is the amplitude. This profile is symmetric to the equator plane and has maximum at the equator for C^ > 0.
As follows from analysis of 1D Parker's equations, which can be derived from Eqs (3), neglecting r-derivatives, solution depends on the product of V = Ca C^, called the dynamo number. Change of the sign of V leads to the change of direction of propagation of the dynamo wave. In general, in 2D case this statement is not correct, and the direction of the wave propagation depends on the spatial distribution of a and Vv. It means that choice of signs of Ca, C^ needs additional information.
The positive sign of a follows from the simple relation a ~ —X between the a-effect and kinetic helicity x, which is negative in the northern hemisphere. From 3D geodynamo simulations follows that Vv has maximum in the bulk of the liquid core at the equator plane [Reshetnyak, 2010]), what is also is in agreement with the helioseismological observations in the solar convective zone [Belvedere et al., 2000]). These two arguments fix the signs of Ca and C^.
Integration in time of Eqs (3)-(4) with a0, and Vv, given by (7)-(8), with the time step t = 10-6, leads to the quasi-periodic oscillatory solution, which has the dipole symmetry for Br and B, and the quadrupole type for Bg, see Figure 1. Note that magnetic field is mostly concentrated inside of the spherical shell, in spite on the penetrating poloidal component of the field outside of the shell. Solution is highly non-linear, what is proved by the very irregular distribution of the a-effect, damped with the magnetic field, see Eq (4).
Evolution of the axi-symmetric magnetic dipole g0, which contributes to the axi-symmetric form of the MauersbergerLowes spectrum [Langel, 1987] Si = (I + 1) (g0) , corresponds to the regime in oscillations, where the mean level of the field is larger rather the amplitude of its fluctuations.
Figure 1. Spatial distribution of Br, Be, B, and a for Ca = 50, = 5104.
The range of oscillations is (0.4 — 0.47), and the dipole does not reverse. This regime is the typical example of the au-
dynamo with the poloidal magnetic energy ^ (B? + B?) of
factor 30 smaller than the toroidal one, B2/2. The ratio of the dipole to quadrupole components Si/S2 ~ 10 is quite large. and remains large for Ca = (5 — 500) for the fixed value of Cw.
To follow the details of the magnetic field generation we consider the butterfly diagrams of the magnetic field, Figure 2. The poloidal field (Br, Be) is taken at the outer boundary, and the toroidal one at the maximum of generation, near the inner boundary. The poloidal field demonstrates two kinds of the waves, propagating to the equator at |0| < 80°, and to the poles at |0| > 70°. Note that there is intersection of the waves in the band 0 = 70 — 80°. Simultaneous existence of the polarwards and equatorwards waves is the subject of debates in the solar dynamo [Moss et al., 2011]. These waves can be related to the quasi-periodic archeomagnetic waves, which also demonstrate different directions of propagation.
The toroidal magnetic field B near the inner boundary oscillates at the non-zero mean level, and at least potentially
Figure 2. The butterfly diagrams for Br, Be at r = ra and B at r = 0.7.
can contribute to the torsional oscillations, concerned with the inner-outer cores interaction. The absolute maximal values of the azimuthal field B in the northern hemisphere is shifted relative to the field in the southern hemisphere at the half of the period of oscillation. It means that solution can not be described with the combination of a few symmetric and antisymmetric functions relative to the equator plane, and that it has more complex structure.
4. Random a
The proposed axi-symmetric aw-model is a crude simplification of the original 3D MHD equations at least in that sense that a0, which describes production of the magnetic field with the turbulence, is a constant parameter. In the more consequent approach [Hoyng, 1993] a0 has a random fluctuating part, caused with the finite number of the fluid cells. This assumption leads to the reasonable estimates of the a fluctuations in the solar convective zone [Moss et al., 2013]. This approach was used to get a spontaneous reversals of the magnetic field in the finite-dimensional geodynamo model [Sobko et al., 2012].
However, we have to use results of the finite-dimensional geodynamo models very carefully because the considered Galerkin decomposition in [Sobko et al., 2012] included only two first modes. On the other hand, input of the energy by the fluctuating a at the small scale can change the magnetic field spectrum essentially. So far there is no inverse cascade in the aw-equations, as it happens, e.g., in 2D hydrodynamic turbulence [Kraichnan and Montgomery, 1980], energy of fluctuations will not transfer over the spectrum to the large scales, and concentrate at the scale of fluctuations. It can happen that such energy injection will lead to the change of the spectrum. In its turn, increase of the energy at the small scales will result in disagreement with observations, which demonstrate predominance of the magnetic dipole component on the higher harmonics. We recall that as it follows from the practice of the 3D dynamo simulations, solution is decently resolved if the kinetic and magnetic energies drop by more than a factor of 100 from the spectral maximum to the cut-off wavelength [Christensen et al., 1999].
Influence of the a fluctuations on the solution of Eqs (3)-(4) was tested, using our finite difference model, which can reproduce continuous spectrum up to the scale of the energy injection. The scale of injection is assumed to be the grid scale, i.e. the distance between the mesh grid points.
The random fluctuations modify a0 in the following way: a0 ^ a0(1 + Cee(r, 6)), where e is the uniformly distributed random variable from —1 to 1, and Ce is the constant. In every mesh grid point e changed after the time St = 0.01 simultaneously, see evolution of the root mean square value of a in Figure 3.
We indeed observed appearance of some reversals of the magnetic field, see evolution of g° in Figure 3, for Ce = 7, 50, which can be related to the geomagnetic field reversals. However, this statement appears to be wrong, because the structure of the magnetic field spectrum Si due to fluctuations changed essentially, see Figure 3. Before it dissipates at the diffusion scale, the magnetic energy of fluctuations accumulates at the wave numbers I > 4, that is resulted in the appearance of the spectrum's plateau at 5 < I < 9. In other words, to change evolution of the magnetic dipole g° one needs to increase the magnetic energy at the small scales in some orders, see the normalized factors for g° in the figure caption. Such a catastrophic event is hardly believed to happen in the liquid core if the geomagnetic reversals is treated like the trivial redistribution of the energy between the harmonics in the white spectrum [Reshetnyak, 2013].
Figure 3. Evolution of the root mean square value of a (upper plane), normalized magnetic dipole g° (middle plane), and the averaged magnetic spectra Si (lower plane) for the three regimes: Ce = 1 (red), 7 (green), and 50 (blue). The corresponding amplitudes for g0: 0.31; 0.32; 1.27; and for Si: 1500; 9900; 124, 000.
5. Geostrophic Regimes
The specific feature of the planetary dynamo is the geo-strophic balance of the forces in the liquid core [Pedlosky, 1987]. Assuming that in the leading order viscous and Archimedean forces are small, one has balance of the Corio-lis force and the gradient of the pressure. Application of the Taylor-Praudman theorem leads immediately to conclusion that velocity field V is elongated along the axis of rotation. In the other words V in the bulk of the core depends weakly on the ^-coordinate.
In the general case, in presence of the viscous force and the thermal buoyancy, locations of the large gradients in the ^-direction correspond to the boundary layers and the equator plane, where physical fields can change the sign. This statement relates not only to the large-scale velocity
6. Dynamic a-quenching
The more sophisticated model of a-quenching is the so-called dynamic quenching, where the damping of a, given by the sum a = a0 + am, is provided with a magnetic part am, described by the evolution equation [Kleeorin et al., 1995]:
-— = B •Vx B - a--—, (10)
at -q T
Figure 4. Spatial distribution of a0 and Vv in the geostrophic regime.
field but to the averaged products of the turbulence, like the kinetic helicity x, a, as well.
Here we use results of 3D simulations of the thermal convection heated from below in the rapidly rotating spherical shell. Roughly, for the moderate Rayleigh numbers (regime R2 in [Reshetnyak, 2010]) a-effect and azimuthal velocity Vv can be approximated as follows:
0.0 = Ca r(-erf (1.25|z|) + 1)e-66.7(s - a39)2 sin(2<9)
= CwS (V11^ - a35)2 + 0.73e-3.§4(s - 1)2) ,
(9)
with the polar coordinates s = r sin 6, z = r cos 6. This approximation corresponds to the convection mainly outside of the Taylor cylinder, see Figure 4.
The maximum of |a01 locates near s = 0.45 and maximum
d V
of the radial gradient of the differential rotation, —--—, is
Or s
near s ~ 0.6, close to |a|'s maximum. It means that the both sources of generation, a-effect, and w-effect have the same locations, and meridional circulation will not change solution too much.
As we can expect from Figure 4, the scale of the magnetic field in s-coordinate is expected to be quite small, because the scales of a-effect, and u are small as well. This prediction is proved with the simulations for Ca = 2980 and C^ = 35.4, which are near the threshold of generation. The ratio of the poloidal to the toroidal energies is equal to 8, and the maximum of the magnetic energy is at I = 3. Increase of Ca, leads to the shift of the maximum of the spectra to the small-scaled part of the spectra. The switch on of the meridional circulation does not help to increase the dipole component of the field.
Magnetic energy oscillates with amplitude about 1% of its mean value, and amplitude of the magnetic dipole oscillations is even smaller. In spite of the fact that we used a and Vv from 3D simulations, production of the dipole magnetic field is less efficient than in the model, discussed in the Section 3.
where T = 1 is the typical time scale. The generated am has the opposite sign to a0 that reduces the total a-effect in (3).
We tested regimes with a0 and Vv, given by (7)-(8), and set of parameters close to Ca = -0.004, C^ = 30. The amplitude of the poloidal magnetic energy (1200) is order of magnitude smaller than the toroidal part (20,000). In spite of the quite large values of the magnetic energies, decrease of Ca, and C& at 20-30% leads to decay of the solution. It can be explained as with the rapid increase of the growth rate in the linearised equations, as well as with coexistence of two finite-amplitude solution branches with the weak and strong magnetic field intensity. Some simulations demonstrate spontaneous transitions from the weak field dynamo to the strong field, accompanied with the reconstruction of the magnetic energy spectrum that tells in favour of the latter assumption. Such a rapid increase of the magnetic field production makes it difficult to find a solution with a predominant dipole contribution.
The magnetic dipole g0 oscillates, changing its sign with the period tosc = 0.036. The magnetic field spectrum has maximum at I = 9, that corresponds to the small-scaled po-larwards dynamo-wave. The further increase of Ca, and C^ preserves the zero mean level of g0. This kind of a-quenching requires a thorough analysis of the range of parameters, which can be used for the geodynamo applications.
7. Conclusions
It is quiet expected that the considered above mean-field dynamo do can reproduce some features of the geomagnetic field. At least in principal, aw-models can generate the predominant dipole magnetic field, similar to that one at the Earth's liquid core, and even the reversals of the field. To the moment it is not clear if the reversal is the intrinsic feature of the dynamo mechanism either it is triggered with the external perturbation. The both scenarios have its own arguments. Here we showed that even the simple idea of the fluctuating mean-field coefficient, say a-effect, should be considered very carefully. Influence of fluctuations on the magnetic dipole evolution should not treated separately from the properties of the magnetic energy spectrum, which can be modified by the fluctuations essentially.
The other point is the application of the 3D dynamo simulations for estimates of the a-effect and differential rotation. Our study reveals that it can not be done straightforward. There are many reasons to that conclusion. One of the reason is that calculation of the averaged quantities like kinetic
helicity and a requires the intermediate physical scale, h, such that Id C h C L, where Id is the dissipative scale, and L is the scale of the liquid core. This is quite difficult task for the 3D simulations, which have still pure resolution. Note also that separation of the scales, well adopted in the astrophysical applications, is questionable point in the geo-dynamo, where the magnetic spectrum is smooth and continuous, and the intermediate scale can absent at all. We also should not exclude possibility that some more successful combination of parameters will improve the situation. This problems requires exploration of the phase space and it is a challenge for the cluster computer systems. It will be the next step of the research in the close future.
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M. Yu. Reshetnyak, Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences, Moscow, Russia. (m.reshetnyak@gmail.com)
