RUSSIAN JOURNAL OF EARTH SCIENCES, VOL. 15, ES4001, doi:10.2205/2015ES000558, 2015
Inverse problem in Parkers dynamo
M. Yu. Reshetnyak1,2
Received 15 November 2015; accepted 17 November 2015; published 19 November 2015.
The inverse solution of the 1D Parker dynamo equations is considered. The method isbased on minimization of the cost-function, which characterize deviation of the modelsolution properties from the desired ones. The output is the latitude distribution of themagnetic field generation sources: the - and -effects. Minimization is made using theMonte-Carlo method. The details of the method, as well as some applications, which can beinteresting for the broad dynamo community, are considered: conditions when the invisiblefor the observer at the surface of the planet toroidal part of the magnetic field is muchlarger than the poloidal counterpart. It is shown that at some particular distributions of and the well-known thesis that sign of the dynamo-number defines equatorial symmetryof the magnetic field to the equator plane, is violated. It is also demonstrated in whatcircumstances magnetic field in the both hemispheres have different properties, and simplephysical explanation of this phenomenon is proposed. KEYWORDS: mean-field dynamo;magnetic field; -, -effects; reversals.
Citation: Reshetnyak, M. Yu. (2015), Inverse problem in Parkers dynamo, Russ. J. Earth. Sci., 15, ES4001,
The observed magnetic field in the various astrophysicalobjects, like planets, stars and galaxies, is a product of thedynamo mechanism. The dynamo theory, which first successwas concerned with the development of the mean-field dy-namo [Krause and Radler, 1980], to the present time trans-formed to the new branch of physics, and combined recentknowledges on the structure and evolution of the objects,fluid dynamics, supercomputer modeling. To now it can de-scribe many typical features of the magnetic field, knownfrom observations [Rudiger et al., 2013], [Roberts and King,2013].
As it usually happens during the development of the newtheory, the first approach is the direct solution of the modelequations with prescribed parameters, which are chosen dueto some a priori information on the system. Whether itleads to the acceptable correspondence of the model with theobservations, the fine tuning of the model parameters starts.This is the subject of the inverse problem, where basing onthe observations, and usually on the fixed equations, the
1Schmidt Institute of Physics of the Earth of the RussianAcademy of Sciences, Moscow, Russia
2Pushkov Institute of Terrestrial Magnetism, Ionosphere andRadio Wave Propagation of the Russian Academy of Sciences,Moscow, Russia
Copyright 2015 by the Geophysical Center RAS.
governing parameters of the model are looked for.There are different ways how it can be done. Here we con-
sider approach, where the desired parameters are the formsof the spatial distribution of the energy sources in the dy-namo equations. We limit our study to the simple, but well-adopted in the dynamo community, 1D Parkers equationswith the algebraic quenching, which are traditionally usedin the planetary, galactic, and stellar dynamo applications[Rudiger et al., 2013]. These equations describe evolutionof the axi-symmetric mean magnetic field, which dependson the latitude . The sources of the energy, the - and-effects, are the prescribed functions of . The aim is tofind such distributions of and in , which satisfy somerestrictions on the simulated magnetic field. The measure ofdeviation of the model from the desired state is characterizedby the cost-function . To minimize numerical expenses wedecompose and in the Fourier series in the polar angle = /2 , and rewrite in terms of the spectral coeffi-cients, where only the first modes are used. Minimizationof , which can have quite complex structure, should bedone using some robust method. So far usually has localminima, we used modification of the Monte-Carlo method,the good candidate for the parallel simulations at the clustersupercomputer systems, used in the work.
Below we consider some examples, which demonstrate im-plementation of the method, and show how information onthe spatial spectrum of the magnetic field, its periodicity,ratio of the poloidal and toroidal magnetic energies can beused for the estimates of the optimal profiles of and . Westress attention that the inverse approach in dynamo appli-
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cations is very rare, compared to the direct simulations, andonly a few papers in this direction exist.
2. Dynamo in the Spherical Shell
We consider simple dynamo model in the spherical shell[Ruzmaikin et al., 1988]:
+ , (1)
where and are the azimuthal components of the vec-tor potential A, and magnetic field B = rotA, () is
the -effect; () is the differential rotation, and =
)is the diffusion operator with
for the magnetic diffusion. System (1) is solved in the in-terval 0 with the boundary conditions = 0 and = 0 at = 0 and .
To exclude the exponentially growing solution of Eqs(1)the -quenching is used. The form of quenching depends onthe particular objects. In planetary and galactic dynamosthe simple algebraic form is acceptable. In the solar dy-namo the dynamical quenching is usually used, see for details[Kleeorin et al., 1995].
Here we consider the local form of the algebraic -quen-ching:
1 + , (2)
with the magnetic energy () = (2 +
2)/2, and radial
component of the magnetic field =1
3. Inverse Problem
The direct solution of the system (1),(2) with the pre-scribed profiles of () and () gives B(, t), which canbe compared with the observations. The disadvantage ofthe direct problem is a pure knowledge on () and ().Thus, in the planetary dynamo these profiles are known onlyfrom 3D simulations, see, e.g., [Reshetnyak, 2010]. For thesolar dynamo [Belvedere et al.,2000] information on comesfrom the helioseismology, however -effect is still varies frommodel to model. In galactic dynamo situation is similar tothe solar dynamo, that is why the simplest models of arestill so popular. These reasons motivate the inverse prob-lem approach, where different profiles of () and () aretested on observations.
Let introduce the cost-function (B, Bo), where B is themodel magnetic field, and Bo is the observable one. Then has at least one minimum at B = Bo. The proper choiceof , and sufficient observations Bo make this minimumglobal. Usually, observations do not cover the whole domainof the magnetic field generation, either one observes such
properties of the magnetic field that magnetic field can notbe recovered in the unique way. Then has local minima aswell, and for minimization of one requires special efficientmethods, see review in [Press et al., 2007].
The next step is to simplify the problem and consideronly the large-scale features of profiles, e.g., the first , Fourier harmonics in :
Then, the problem reduces to the search of such C andC that (C, C) has minimum (maybe local). In gen-eral case, study of the sequence of minima, obtained duringsimulations, is interesting too.
The numerical details of the direct solver, based on thecentral 2-order finite-difference approximation of the spa-tial derivatives, and 4-order Runge-Kutta method for in-tegration in time, are described in [Reshetnyak, 2014]. Thedirect C++ solver was wrapped, using MPI interface, sothat at each computer node the direct problem (1),(2) forthe different (C, C), given by the random generator, wassolved.
The random Gauss generator, with the mean value, equalto the previous best choice, and standard deviation 3, gen-erates set of (C, C). It is supposed that (C, C) shouldbe in the fixed region. After selection of (C, C) at thecurrent iteration step, which corresponds to the minimal ,the new (C, C) were generated, and then the next itera-tion started. The shift of the mean value of (C, C), whichis optional, helps to increase convergence of the process.This method is modification of the Monte-Carlo method,see the basic ideas in [Press et al., 2007].
To solve equations at Lomonosovs supercomputer in Mos-cow State University and the Joint Supercomputer Center ofRAS, = 101 grid points for the spatial approximation, thetime step = 105, and computer nodes from 10 to 100for parallelization were used. Usually, number of iterationswas less than 10, and depended on . Application of MPIand cluster computers for 1D problem is not crucial, butit will be of great importance for the 2D code (with radialdependence), which is under development now.
Further we consider some particular forms of and dis-cuss the resulted profiles of () and () in details.
4. Ratio of the Poloidal and ToroidalMagnetic Energies
The measure of intensity of generation sources in (1) is the
so-called dynamo-number, defined as: = |||| ||||3
where = is the spatial scale, and ||.|| is the norm of thephysical quantity, discussed below. Here we consider howsolution of (1),(2), with fixed |||| and ||||, depends onthe forms of profiles.
Having in mind that the both quantities , , can changethe sign, we introduce the following definitions of norms:
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Figure 1. Latitude dependence of , , and their product for minimal .
|||| = 1
|| sin , and |||| = 1
|| sin . It
would correspond to the classical definition of the dynamonumber with the fixed amplitudes of and for the uniformprofiles.
We look for such solutions of (1), (2), which for the fixed||||, ||||, have minimal, either maximal ratio of thepoloidal =
2/2 and toroidal = 2/2 magnetic
energies.So far in many astrophysical applications only the one
component of the magnetic field (poloidal or toroidal) canbe observed, estimate of the whole magnetic energy =(1+) can vary from model to model, and amplitude ofits variations is the subject of active debates [Brandenburgand Subramanian, 2005].
Figure 2. Latitude dependence of , , and their product for maximal .
Simple analysis of (1),(2) leads to the following predic-tions:
||||24 , 1,
that follows to that is defined by |||| and ||||. Our aimis to find dependence of on these profiles.
Let introduce the cost-function = 1 , and find(C, C), which extremum of . Latitude distributions of and for the four cases = = with = 2 . . . 5,and |||| = 102, |||| = 102, are presented in Figure 1 andFigure 2.
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Firstly note that some details do depend on . Thisis natural for the small . However usage of the large would be inconsistent with the basics of the mean-field dy-namo, where the large-scale fields are considered. In otherwords, the number of harmonics should be much less thanthe number of the grid points in the numerical scheme forEqs(1). It means that here we discuss only the large-scaletrends in the model, and they do exist.
Before to start the analysis of Figure 1 and Figure 2,note that minimal ( 104) and maximal 1correspond to the different levels of the total magnetic energy: for one has 103, and for 1.In agreement with estimate (3), the case correspondsto 1. On contrary, in the case , by some reasons,there is suppression of the total magnetic field generation.
Following further note that due to our normalization, am-plitudes of , and , see Figure 1 and Figure 2, do notdemonstrate significant differences. But as was already men-tioned before, the measure of the field generation is the prod-uct . And this quantity does demonstrate the differentbehaviour for two branches. For (large ) there isonly one extremum of in the hemisphere. This helps togenerate the large-scale magnetic field.
For (small ) the product oscillates in co-ordinate. The scale of the fields is smaller than in the caseof , and as a result, the magnetic diffusion is larger.Whether for for all , the leading harmonic for isstable quadrupole (Legendre polynomial with = 2), thenfor during the time solution switches from = 1(dipole) to higher orders: even to = 10 at = 5. Sofar the amplitudes of and are of the same order in theboth cases, difference in is a product of low correlation inspace of and , as well as of the energy sources with thegenerated magnetic field. The first option is shown in Fig-ure 1, where the maximum of the product near the equatorialplane is clearly pronounced. On contrary, this correlation issmall in Figure 2. It supports suggestion that localizationsof the both energy sources ( and ) in the same placehelps to the large-scale magnetic field generation.
The test on the field configurations reveals that for the both components of the quadrupole magnetic field havemaximum at the equator, so that in that region the productsof the magnetic field components and , are large, andas a result, the magnetic field generation is enhanced.
For the case correlation between the magnetic fieldand energy sources is weak, and efficiency of the dynamomechanism is small. Situation can change if the meridionalcirculation, providing transfer of the magnetic field from oneregion of generation to the other, will be taken into account.Then effective generation of the magnetic field with the dif-ferent localizations of and is possible.
5. Pure Dipole and Non-dipole Solutions
The another prediction of the linear analysis of Parkerequations with simple forms of and is that alternationof sign leads to the change of the symmetry of the leadingmode: the dipole mode switches to the quadrupole, and vice
versa. This change can also be accompanied with transitionfrom the stationary to oscillatory regimes. Using our ap-proach we test whether this prediction is valid for complexforms of and in the non-linear regime.
Let introduce the cost-function = 121/
are the spectral coefficients in decomposition on the Leg-endre polynomials. The same norms |||| and ||||, as inthe previous section, were used. Minimum of correspondsto the non-dipole configuration, and maximum limits to thepure dipole field, respectively. As we will see, the two groupswith dipole ( = 1) and non-dipole ( > 1) configurations willdominate.
The four runs with = 2 . . . 5 for minimal and maximal were done. For maximal the stationary dipole solutionwas observed for all the runs. The toroidal energy was 650, and the poloidal one was two orders less. Exception wasthe case with = 5 with 100, and 0.1.
The regimes with minimal demonstrated various be-haviour in time. Cases with = 2, 4 were the stationaryquadrupoles with 0.1 and 0.01, and 900, 40, re-spectively. In the case = 3 we got 1, 10. Thedominant oscillatory mode was = 6. The last stationaryregime with = 5 corresponded to = 4.
The visual analysis of product does not reveal anysignificant differences between the branches of the minimaland maximal . To test whether the sign plays the role,
we calculated integrals/20
, for : 4.3 104, 2.7 104,
-4 103, -2 104, and for : 3.4 104, 3.2 104, 2.1 104, -5 103.As we can see, the sign of the integral does not influenceon whether solution is dipole, either it is quadrupole. More-over, there is no correlation of sign of with the symmetryof the magnetic field over the equator plane in the non-linearregime. This result demonstrates once more how predictionsof the linear analysis should be used carefully in the satu-rated states.
6. Dynamo-wave Through Equator
The asymmetry of the magnetic fields over the equatorplane is well-known to observers. In geomagnetism thisproblem was discussed in [Gubbins et al., 2000], where theidea of the interplay of the dipole and quadrupole modeswas proposed. These two modes have similar thresholds ofgeneration and its superposition can enforce the total mag-netic field in one hemisphere, and weaken it in the other one.The paleomagnetic records, often based on the assumptionof the axial dipole, can not exclude this possibility even forPhanerozoic.
In the solar dynamo asymmetry presents at least in twoforms: the difference between the magnetic fluxes fromtwo hemispheres is finite, and can change the sign in time[Knaack et al., 2004]. The other remarkable phenomenonis that during the Maunder minimum in the 17 centurymore than 95% of the sunspots were located in the southernhemisphere of the Sun [Ribes and Nesme-Ribes, 1993].
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Another example of the break of the magnetic field equa-torial symmetry demonstrates Marss crustal field [Stanley etal., 2008]. This field is associated with the internal magneticfield generated by the dynamo mechanism in the past.
The equatorial asymmetry of the magnetic field is allowedby the dynamo theory as well. The 3D dynamo simulationscan reproduce this phenomenon for the particular set of pa-rameters as for the spherically symmetrical boundary con-ditions [Grote and Busse, 2000], [Busse and Simitev, 2006],[Landeau and Aubert, 2011], as well as for the heterogeneousheat flux at the outer boundary of the spherical shell [Stan-ley et al., 2008], [Amit et al., 2011], [Dietrich and Wicht,2013].
It should be noted that possibility of such asymmetries isalso interesting from the general point of view. It motivatesus to use the inverse approach to test this phenomenon atthe simple dynamo-model.
In assumption that dynamo wave, say for the field , ismonochromatic, its phase velocity is =
tion on can be used to distinguish between the two cases:the wave, which propagates through the equator plane, ei-ther it vanishes at the plane, and then recovers with theopposite sign in the second hemisphere2.
The mean value of in the equatorial band = is
. In assumption, that the band is narrow
enough, so that changes (if does) the sign only at the
equator, the normalized quantity = /
in the interval [0, 1]. The case = 0 corresponds to thevanishing wave at the plane = 0. The second extreme caseis || = 1, when has the same sign over the whole band.
The proposed cost-function has the following form:
The first term in the sum in (4) corresponds to the de-scribed above restriction on the wave behaviour in the band.The second term helps to filter out the non-oscillatory solu-tions:
= 11 + 2
, 1 = , 2 = ,
where the overline means averaging over the whole spaceand time. The case with 1 corresponds to the smallamplitude oscillations, compared to the mean level of .We do not interesting in this regime. The case with 1corresponds to the large oscillations: e.g., for = sin
2(),and any integer , 0.68.
The largest and provide minimum of in (4).The simulated magnetic field, see the butterfly diagrams
in Figure 3, demonstrate the quite different behaviour inthe northern and southern hemispheres. In the northernhemisphere it consists of two kinds of waves, which travelto the poles at the high latitudes, and from the poles to theequator in the band = /4. There are periodic reversals
2The third possibility, when the wave is reflected from theequator plane, is not supported by the observations.
Figure 3. The butterfly diagrams for , componentsof the magnetic field, and phase velocity of the poloidalmagnetic field for = 2 and |||| = ||| = 50.
of the magnetic field, which correspond to the change of thesign of .
On the contrary, in the southern hemisphere the mainpart of the magnetic field is constant in time. The poloidalfield is concentrated near the pole, and maximum of thetoroidal field is shifted to /3.
This quite strange configuration of the magnetic field, at
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Figure 4. Latitude dependence of , , and their product with = 2.
least compared to the usual field states, corresponds to theclass of the hemispherical dynamo, mentioned in the begin-ning of the section. Note that we did not use any imposedasymmetry in the model, and this result is the intrinsic prop-erty of the model, as it was discovered in some 3D simula-tions.
Returning to the way how we get this solution, we remindthat the crucial point was selection of regimes with the non-zero mean phase velocity of the radial magnetic field inthe equatorial band, see Figure 3. There are waves of ,traveling from the north pole to the southern, with the con-stant magnitude, except the region near the equator plane,where is small. If resolution of observations is pure, thenit seems that the dynamo wave penetrates free through theequator plane from one hemisphere to the other. The direc-tion of this wave changes in time, however the mean value of over the time and space domain in Figure 3 is not zero.It is this deviation from the zero value the cost-function (4)detected.
The possible explanation of our hemispherical dynamo isconcerned with the spatial distribution of and , obtainedin the inverse model, Figure 4.
We observe coincidence of and extremas locations.It results in the large product . Situation is similar tothat one in Figure 1, where correlation of and wasalso strong. However, in that case extrema of were nearthe equator plane, on contrary to the hemispherical dynamo,where they are shifted to the middle latitudes. It is this shiftof maximum of the magnetic field generation helps to isolatedynamo process in hemispheres from each other, and permitsdifferent evolutions of the magnetic field in the hemispheres.We emphasize that the observed flux, concerned with thephase velocity , is quite small, and does not change sit-uation substantially. But as we demonstrated, this flux isthe result of the equatorial symmetry break, which leads tothe very different morphologies of the magnetic fields in thehemispheres.
Having deal with the direct dynamo problem solutions, Ireally enjoyed to work with the inverse problem approach forthis toy dynamo model. In spite of the fact that 1D modelitself is out of date, the level of abstraction in communi-cation with the computer in the inverse approach is muchhigher than in the direct problems. In the inverse approachone formulates the properties of the desired solution, andthen tries to understand why the resulted parameters pro-vide these properties. This process is much more intriguingrather than to use the fixed parameters, and follow the re-sults of the direct problem, where solution is already alsoprescribed. However the latter approach can be used for themore sophisticated models, its not the fact that the simplermodel in inverse approach will not give the better result dueto the finer tuning of parameters.
The obtained above results are the product of numeroustries, when for many times I wandered why the computerselected this or that particular regime. The lack of criteria,which were used for the cost-function construction, some-times resulted in the very unexpected results. Many restric-tions, which are supposed by default, should be explainedstraightforward to the computer. However the results areworthy of these efforts. May be what is more important, isthat this approach stimulates understanding of the model.With minimal number of criteria, it is possible to find scenar-ios, which can be tested, using more complex models. Thisinverse approach can be useful tool for asking a good ques-tions, even the answers would be quite wrong. As regardsto the simplicity of the considered model, estimates of therequired computer time shows that the inverse method, con-sidered here, can be extrapolated to the higher dimensionalmodels as well.
Acknowledgment. The author is grateful to I. Aleshin for
stimulating discussions. The author also acknowledges financial
support from RFBR under grants 15-05-00643, 15-52-53125.
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M. Yu. Reshetnyak, Schmidt Institute of Physics of the Earthof the Russian Academy of Sciences, Moscow, Russia,
Pushkov Institute of Terrestrial Magnetism, Ionosphere andRadio Wave Propagation of the Russian Academy of Sciences,Moscow, Russia, ([email protected])
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Abstract1. Introduction2. Dynamo in the Spherical Shell3. Inverse Problem4. Ratio of the Poloidal and Toroidal Magnetic Energies5. Pure Dipole and Non-dipole Solutions6. Dynamo-wave Through Equator7. ConclusionsReferences
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