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open questions in stellar physics led to the idea that a dynamo operates in the radiative cores of early - type stars ( spruit 2002 ) .
even the helioseismologic observation of rigid rotation of the solar interior shows in this direction .
the angular momentum transport by the large - scale magnetic field pattern ( fossil field plus toroidal field induced by differential rotation ) does _ not _ lead to a solid - body rotation unless the viscosity of the plasma exceeds the molecular value by a few orders of magnitude ( rdiger & kitchatinov 1996 , eggenberger et al .
other examples are given by the evolution of the fast rotating early - type stars which can only be understood if i ) there is a basic transport of angular momentum outwards and ii ) the radial mixing of chemicals remains weak ( yoon et al .
2006 , brott et al .
2008 ) . hence ,
if a ( magnetic - induced ) instability existed in the radiative stellar cores then the corresponding schmidt number @xmath3 must be rather large .
we have shown that the kink - type instability ( or tayler instability , ti ) of toroidal magnetic fields forms a much - promising candidate for the instability . a schmidt number larger than ten results as the ratio of the effective viscosity and the diffusion coefficient ( rdiger et al .
2009 ) .
the unstable modes of the ti are basically nonaxisymmetric driven by the energy of the electrical current which produces the toroidal field .
interestingly enough , there exists even an instability of a toroidal magnetic field which in the fluid is current - free . in this case
the energy comes from a differential rotation which itself is stable but which is unstable under the influence of the ( current - free ) toroidal field .
we have called this instability as azimuthal magnetorotational instability ( amri ) as  like for the standard mri ( with axial fields )  the field itself is current - free and does not exert forces . in opposition to the standard mri
the amri is always nonaxisymmetric and it is , therefore , much more interesting for the dynamo theory . for complicated radial profiles of the toroidal field we shall always have a mixture of ti and amri .
generally , the latter is more important for fast rotation ( @xmath4 ) and v.v . here
the alfvn frequency @xmath5 for the toroidal field is used which derives from the alfvn velocity @xmath6 as the related frequency . between two cylinders with different radii
the toroidal field profile with @xmath7 ( @xmath8 radius ) is free of dissipation .
the ` perfect ' amri appears for @xmath9 while the ` perfect ' ti results for @xmath10 .
one can compute the necessary electrical currents to excite both sorts of instabilities in a columnar taylor - couette experiment with gallium as fluid conductor .
as we have shown the critical hartmann numbers for self - excitation of axi- and nonaxisymmetric perturbation modes do not depend on the magnetic prandtl number of the fluid which is as small as @xmath11 for stellar plasma and @xmath12 for liquid sodium ( rdiger & schultz 2010 ) .
it is typical for the nonaxisymmetric ti and amri that always the two modes with @xmath13 are excited for the same critical hartmann number and also  if supercritical  with the same growth rates ( fig .
[ fig1 ] ) . despite their simultaneous existence
they can be excited as singles with different initial conditions . however , if the initial conditions are as neutral as possible with respect to a preferred helicity , in the majority of the cases one of the modes dominates after our experiences .
this behavior may have dramatic consequences with respect to the dynamo theory .
both the modes with @xmath13 have opposite helicity with the same total amount .
the mode with @xmath14 is identical to the mode with @xmath15 but in a left - hand system .
the helicity of @xmath15 in the right - hand system equals the helicity of @xmath14 in the left - hand system .
so it is obvious that in one and the same coordinate system the sum of the helicity of @xmath14 and @xmath15 is zero . as a consequence
the instability of a toroidal field can only develop helicity if by some reasons one of the modes with @xmath13 dominates the other .
there are the two possibilities that i ) one mode dominates the other by chance ( so as the matter dominates the antimatter ) or ii ) the existence of a poloidal field prefers one of the modes .
we have shown that indeed in stellar radiation zones  if the background field has a positive current helicity @xmath16  the resulting kinetic helicity @xmath17 of the fluctuations is always negative ( gellert et al .
a positive current helicity of the background field results if an axial field and an axial electrical current are parallel . a negative current helicity of the background field results if an axial field and an axial electric current are antiparallel .
hence , the resulting kinetic helicities on the basis of current - driven instabilities have , therefore , opposite signs in opposite hemispheres of the model .
we are interested in the stability of a background field @xmath18 with @xmath19 , and the flow @xmath20 in a dissipative conducting fluid rotating between two rigid cylinders .
@xmath21 is the kinematic viscosity and @xmath22 is the magnetic diffusivity , their ratio is the magnetic prandtl number @xmath23 .
the stationary background rotation law is @xmath24 with @xmath25 and @xmath26 as constants . @xmath27 and @xmath28 are the rotation rates of the cylinders and @xmath29 and @xmath30 are the azimuthal magnetic fields there .
one finds for the current helicity of the background field @xmath31 which may be either positive or negative ( and of course vanishes for the current - free case @xmath9 ) .
the inner value @xmath32 may be normalized with the uniform vertical field , i.e. @xmath33 for a profile with @xmath34 we have @xmath35 for the normalized current helicity of the background field .
the sign of @xmath36 determines the sign of the current helicity . as usual , the toroidal field amplitude is measured by the hartmann number @xmath37 and the global rotation by the reynolds number @xmath38 with @xmath39 .
the alfvn frequency is @xmath40 .
the boundary conditions associated with the perturbation equations are no - slip for the flows and perfectly conducting for the fields . for all computations
it is @xmath41 .    for purely toroidal fields
the expected net helicities vanish as two nonaxisymmetric modes with @xmath13 exist with identical excitation conditions and the same amount of helicity but of opposite sign ( fig .
[ fig1 ] ) .     the normalized negative kinetic helicity and the negative current helicity for the nonaxisymmetric perturbations as functions of @xmath36 .
@xmath42 , @xmath43 , @xmath44 , @xmath45.,width=302 ]    we now compute both the helicities for helical fields with various pitch values @xmath36 which is inverse to the current helicity of the large - scale field . the astrophysically relevant case is @xmath46 ( which is not the classical realization of the tayler instability ) .
the instability only exists for nonrigid rotation so that we always work with a rotation law @xmath47 .
the radial profile of the toroidal field has been modelled by the above mentioned most simple profile with almost uniform toroidal field .
figure [ fig2 ] gives the results ( in units of @xmath48 ) .
both the kinetic helicity @xmath17 as well as the current helicity @xmath49 are negative .
they are decreasing functions of @xmath36 .
we have also calculated the @xmath0-effect via the determination of the electromotive force @xmath50 by the fluctuations . according to the rule that the azimuthal @xmath0-effect is anti - correlated with the ( kinetic ) helicity we expect the azimuthal @xmath0-effect as positive for @xmath51 .
this is indeed the case ( see fig .
[ fig3 ] , left ) .
one finds positive and negative signs in the container but the positive values dominate so that in the average the azimuthal @xmath0-effect is indeed positive .
on the other hand , for negative @xmath36 we expect positive small - scale kinetic helicity and negative @xmath0-effect .
the example presented in fig .
[ fig3 ] refers to the pitch @xmath52 . for higher values , i.e. smaller helicity of the background field ,
the resulting @xmath0-effect becomes smaller and smaller .
figure [ fig3 ] ( right ) shows that the dynamo number @xmath53 runs as @xmath54 with @xmath55 of order 10@xmath56 .
it is easy to show that this value is too small by two orders of magnitudes to allow the operation of a classical @xmath57-dynamo ( gellert et al .
we start with the amplification of fossil magnetic fields by shear and the magnetic - field back - reaction .
the two - dimensional , non - linear simulations thus start with an initial differential rotation and a purely poloidal magnetic field in a radiative stellar zone .
the early phase of the simulation shows a generation and steep amplification of toroidal magnetic field through differential rotation .
the generated lorentz force redistributes the angular momentum .
this is why at the same time of field amplification , the differential rotation starts to decrease , and the toroidal - field growth is thus limited .
figure  [ fig4 ] shows the maximum magnetic field amplitude in the spherical shell as a function of time .
the reynolds number is @xmath58 and the magnetic prandtl number is @xmath45 .     the stability limits of each snapshot of the axisymmetric evolution of a sufficiently strong initial field .
the solid line gives the evolution of the toroidal field .
the stability limit is given for the nonaxisymmetric perturbation patterns s1 and a1 ( from arlt & rdiger 2011).,width=257 ]    the dotted and dot - dashed lines in fig .
[ fig4 ] are the results of a linear stability analysis of only the toroidal field . by s1
we refer to a velocity perturbation which is symmetric with respect to the equator and has @xmath15 , a1 is the corresponding antisymmetric perturbation . if at any given time both stability lines are above the solid line , the corresponding snapshot is stable against @xmath15 perturbations .
the stability lines cross the solid one at about @xmath59 ( see arlt & rdiger 2011 ) .
the system is now perturbed in a 3d , nonlinear simulation at a somewhat later time ( @xmath60 ) .
the perturbation is applied to the magnetic field and has an azimuthal wave number of @xmath15 ( and is symmetric with respect to the equator ) .
the resulting flow is thus antisymmetric and can be compared with the a1-mode in the linear stability diagram in fig .
[ fig4 ] .    the spatial distribution of the resulting helicity is shown in fig .
note the concentration of both the helicities near the inner boundary where the tangent cylinder touches the inner sphere .
we conclude that a considerable part of the kinetic helicity in the system is due to the presence of an inner sphere .
this is slightly different for the current helicity . a considerable amount of _ positive _ current helicity is measured in the bulk of the northern hemisphere .
the situation is unchanged in runs with perfect conductor boundary conditions at the inner radius . interestingly ,
reshetnyak ( 2006 ) with convection simulations also finds negative kinetic helicity along the tangent cylinder .
we assume that the negative helicity near the tangent cylinder of the northern hemisphere is an inner - boundary effect , neither related to convective nor tayler instability turbulence .    .
middle : kinetic helicity @xmath61 .
right : the azimuthal component of the @xmath0-effect .
, width=408,height=264 ]    figure [ fig6 ] summarizes the consequences of a numerical experiment . in the unstable domain of fig .
[ fig4 ] the instability pattern is calculated without global rotation , i.e. for @xmath63 . in this case
the only existing pseudo - scalar is @xmath1 like in the above theory in cylindric geometry .
the question is whether we can find the same relations between large - scale current helicity ( left ) , kinetic helicity ( middle ) and @xmath0-effect ( right ) .
note first that indeed the dominant role of the tangent cylinder for the kinetic helicity and the @xmath0-effect vanishes so that it is obvious that it is a rotation - induced boundary layer effect . in the southern hemisphere
the pseudo - scalar @xmath1 is positive while the kinetic helicity is negative ( but concentrated to the pole ) .
the corresponding @xmath0-effect given in the right panel ( and computed also by means of the test - field method ) proves to be positive ( see arlt & rdiger 2011 ) . as it must
, at the northern hemisphere all signs are opposite .
hence , the model fulfills the same sign rules as in the above - discussed cylindric setup .
unstable toroidal fields alone are not able to produce helicity and @xmath0-effect .
it has been shown , however , that helicity and @xmath0-effect are produced by unstable magnetic large - scale field patterns which themselves possess current helicity @xmath1 .
this is insofar understandable as helicity and @xmath0-effect are both pseudo - scalars which can only be nonvanishing if in the global setup a nonvanishing large - scale pseudo - scalar like @xmath1 exists .
we want to stress , however , that the current helicity of the background field is not the only possible pseudo - scalar existing in magnetized stellar radiation zones on which other forms of helicity and @xmath0-effect may base .
it is shown that the magnetic current - driven ( ` kink - type ' ) instability produces flow and field patterns with helicity and even with @xmath0-effect but only if the magnetic background field possesses non - vanishing current helicity @xmath1 by itself . 
 fields with positive large - scale current helicity lead to negative small - scale kinetic helicity . 
 the resulting @xmath0-effect is positive . 
 these results are very strict for cylindric setups without @xmath2-dependence of the background fields . 
 the sign rules also hold for the more complicated cases in spheres where the toroidal fields are the result of the action of differential rotation ( induced from fossil poloidal fields ) at least for the case that the global rotation is switched off after the onset of the instability .