Budget equations and astrophysical nonlinear mean-field dynamos

Solar, stellar and galactic large-scale magnetic fields are originated due to a combined action of non-uniform (differential) rotation and helical motions of plasma via mean-field dynamos. Usually, nonlinear mean-field dynamo theories take into account algebraic and dynamic quenching of alpha effect and algebraic quenching of turbulent magnetic diffusivity. However, the theories of the algebraic quenching do not take into account the effect of modification of the source of turbulence by the growing large-scale magnetic field. This phenomenon is due to the dissipation of the strong large-scale magnetic field resulting in an increase of the total turbulent energy. This effect has been studied using the budget equation for the total turbulent energy (which takes into account the feedback of the generated large-scale magnetic field on the background turbulence) for (i) a forced turbulence, (ii) a shear-produced turbulence and (iii) a convective turbulence. As the result of this effect, a nonlinear dynamo number decreases with increase of the large-scale magnetic field, so that that the mean-field $\alpha\Omega$, $\alpha^2$ and $\alpha^2\Omega$ dynamo instabilities are always saturated by the strong large-scale magnetic field.


INTRODUCTION
Large-scale magnetic fields in the Sun, stars and galaxies are believed to be generated by a joint action of a differential rotation and helical motions of plasma (see, e.g., Moffatt 1978;Parker 1979;Krause & Rädler 1980;Zeldovich et al. 1983;Ruzmaikin et al. 1988;Rüdiger et al. 2013;Moffatt & Dormy 2019;Rogachevskii 2021;Shukurov & Subramanian 2021).This mechanism can be described by the αΩ or α 2 Ω mean-field dynamos.In particular, the effect of turbulence in the mean-field induction equation is determined by the turbulent electromotive force, u × b , which can be written for a weak mean magnetic field B as u × b = α K B + V (eff) × B − η T (∇ × B), where α K is the kinetic α effect caused by helical motions of plasma, η T is the turbulent magnetic diffusion coefficient, V (eff) is the effective pumping velocity caused by an inhomogeneity of turbulence.Here the angular brackets imply ensemble averaging, u and b are fluctuations of velocity and magnetic fields, respectively.The threshold of the αΩ mean-field dynamo instability is described in terms of a dynamo number DL = α K δΩ L 3 /η 2 T , where δΩ characterises the non-uniform (differential) rotation and L is the stellar radius or the thickness of the galactic disk.
The mean-field dynamos are saturated by nonlinear effects.In particular, a feedback of the growing large-scale magnetic field on plasma motions is described by algebraic quenching of the α effect, turbulent magnetic diffusion, and the effective pumping velocity.This implies that the turbulent transport coefficients, α K B , η T B and V (eff) B depend on the mean magnetic field B via algebraic decreasing functions.The quantitative theories of the algebraic nonlinearities of the α effect, the turbulent magnetic diffusion and the effective pumping velocity have been developed using the quasi-linear approach for small fluid and magnetic Reynolds numbers (Rüdiger & Kichatinov 1993;Kitchatinov et al. 1994;Rüdiger et al. 2013) and the tau approach for large fluid and magnetic Reynolds numbers (Field et al. 1999;Rogachevskii & Kleeorin 2000, 2001, 2004, 2006).
In addition to the algebraic nonlinearity, there is also a dynamic nonlinearity caused by an evolution of magnetic helicity density of a small-scale turbulent magnetic field during the nonlinear stage of the mean-field dynamo.Indeed, the α effect has contributions from the kinetic α effect, α K , determined by the kinetic helicity and a magnetic α effect, α M , described by the current helicity of the small-scale mag-netic field (Pouquet et al. 1976).The dynamics of the current helicity are determined by the evolution of the smallscale magnetic helicity density Hm = a•b , where b = ∇×a and a are fluctuations of the magnetic vector potential.The total magnetic helicity, i.e., the sum of the magnetic helicity densities of the large-scale and small-scale magnetic fields, HM + Hm, integrated over the volume, (HM + Hm) dr 3 , is conserved for very small microscopic magnetic diffusivity η.Here HM = A•B is the magnetic helicity density of the large-scale magnetic field B = ∇×A and A is the mean magnetic vector potential.
As the mean-field dynamo instability amplifies the mean magnetic field, the large-scale magnetic helicity density HM grows in time.Since the total magnetic helicity (HM + Hm) dr 3 is conserved for very small magnetic diffusivity, the magnetic helicity density Hm of the small-scale field changes during the dynamo action, and its evolution is determined by the dynamic equation (Kleeorin & Ruzmaikin 1982;Zeldovich et al. 1983;Gruzinov & Diamond 1994;Kleeorin et al. 1995;Kleeorin & Rogachevskii 1999), which includes the source terms and turbulent fluxes of magnetic helicity (Kleeorin & Rogachevskii 1999;Kleeorin et al. 2000;Blackman & Field 2000 Taking into account turbulent fluxes of the smallscale magnetic helicity, it has been shown by numerical simulations that a nonlinear galactic dynamo governed by a dynamic equation for the magnetic helicity density Hm of a small-scale field (the dynamical nonlinearity) saturates at a mean magnetic field comparable with the equipartition magnetic field (see, e.g., Kleeorin et al. 2000Kleeorin et al. , 2002Kleeorin et al. , 2003a;;Blackman & Brandenburg 2002;Brandenburg & Subramanian 2005;Shukurov et al. 2006;Chamandy et al. 2014;Chamandy & Singh 2018).Numerical simulations demonstrate that the dynamics of magnetic helicity plays a crucial role in solar and stellar dynamos as well (see, e.g., Kleeorin et al. 2003bKleeorin et al. , 2016Kleeorin et al. , 2020Kleeorin et al. , 2023;;Sokoloff et al. 2006;Zhang et al. 2006Zhang et al. , 2012;;Käpylä et al. 2010;Hubbard & Brandenburg 2012;Del Sordo et al. 2013;Safiullin et al. 2018;Rincon 2021).Different forms of magnetic helicity fluxes have been suggested in various studies using phenomenological arguments (Kleeorin & Rogachevskii 1999;Kleeorin et al. 2000Kleeorin et al. , 2002;;Vishniac & Cho 2001;Subramanian & Brandenburg 2004;Brandenburg & Subramanian 2005).Recently, the turbulent magnetic helicity fluxes have been rigorously derived (Kleeorin & Rogachevskii 2022;Gopalakrishnan & Subramanian 2023).In particular, Kleeorin & Rogachevskii (2022) apply the mean-field theory, adopt the Coulomb gauge and consider a strongly density-stratified turbulence.They have found that the turbulent magnetic helicity fluxes depend on the mean magnetic field energy, and include non-gradient and gradient contributions.In addition, Gopalakrishnan & Subramanian (2023) have recently shown that contributions to the turbulent magnetic helicity fluxes from the third-order moments can be described using the turbulent diffusion approximation.
In a nonlinear αΩ dynamo one can define a nonlinear If the nonlinear dynamo number DN B decreases with the increase of the large-scale magnetic field, the mean-field dynamo instability is saturated by the nonlinear effects.However, if the α effect and the turbulent magnetic diffusion are quenched as (B/Beq) −2 for strong mean magnetic fields, the nonlinear dynamo number DN B ∝ (B/Beq) 2 increases with the increase of the large-scale magnetic field, and the mean-field dynamo instability cannot be saturated for a strong mean magnetic field.Here Beq = µ0 ρ u 2 1/2 is the equipartition mean magnetic field and µ0 is the magnetic permeability of the fluid.How is it possible to resolve this paradox?
The mean-field dynamo theories of the algebraic quenching imply that there is a background helical turbulence with a zero mean magnetic field.The large-scale magnetic field is amplified by the mean-field dynamo instability.In a nonlinear dynamo stage, the dissipation of the generated strong large-scale magnetic field results in an increase of the turbulent kinetic energy of the background turbulence.The latter effect causes an increase of the turbulent magnetic diffusion coefficient and decrease of the nonlinear dynamo number.This additional nonlinear effect results in a saturation of the dynamo growth of a strong large-scale magnetic field.
However, this nonlinear effect has not been yet taken into account in nonlinear mean-field dynamo theories which derive the algebraic quenching of the turbulent magnetic diffusion.In the present study, we have taken into account this feedback effect of the mean magnetic field on the background turbulence using the budget equation for the total (kinetic plus magnetic) turbulent energy.Considering three different types of astrophysical turbulence: • a forced turbulence (e.g., caused by supernova explosions in galaxies); • a shear-produced turbulence (e.g., in the atmosphere of the Earth or other planets) and • a convective turbulence (e.g., in a solar and stellar convective zones), we have demonstrated that the nonlinear dynamo number indeed decreases with the increase of the mean magnetic field for any strong values of the magnetic field, resulting in saturation of the mean-field dynamo instability.
This paper is organized as follows.In Sec. 2 we explain the essence of the algebraic and dynamic nonlinearities, and discuss the procedure and assumptions for the derivation of the nonlinear turbulent electromotive force (EMF).In Sec. 3 we consider the budget equations for the turbulent kinetic and magnetic energies which allow us to take into account the increase of turbulent kinetic energy of the background turbulence by the dissipation of a strong mean magnetic field and to determine asymptotic properties of turbulent magnetic diffusion and nonlinear dynamo numbers for a strong mean magnetic field for the mean-field α Ω dynamo (see Sec. 4), the α 2 dynamo (see Sec. 5) and the α 2 Ω dynamo (see Sec. 6).In addition, in Sec. 5 we discuss a long-standing question when a kinematic α 2 dynamo can be oscillatory, and in Sec.6 we outline important asymptotic properties in the α 2 Ω dynamo.Finally, in Sec.7 we discuss the obtained results.
To explain the essence of the algebraic and dynamic nonlinearities, we discuss in this section the procedure and assumptions for the derivation of the nonlinear turbulent electromotive force in a non-rotating and helical small-scale turbulence.In the framework of the mean-field approach, the mean magnetic field B is determined by the induction equation where U is the mean velocity (differential rotation), η is the magnetic diffusion due to the electrical conductivity of plasma and E(B) = u × b is the the turbulent electromotive force (EMF).To derive equations for the nonlinear coefficients defining the turbulent electromotive force (EMF), we use a mean-field approach in which the magnetic and velocity fields, the fluid pressure and density are separated into the mean and fluctuating parts, where the fluctuating parts have zero mean values.We consider the case of large hydrodynamic and magnetic Reynolds numbers.The momentum and induction equations for the turbulent fields are given by where ρ is the mean plasma density, µ0 is the magnetic permeability of the plasma, F is a random external stirring force, u N and b N are the nonlinear terms which include the molecular dissipative terms, ptot = p + (µ0 ρ ) −1 (B • b) are fluctuations of the total pressure and p are fluctuations of the plasma pressure.For simplicity, let us consider incompressible flow, so that the velocity u satisfies to the continuity equation, ∇ • u = 0 and the fluid density is constant.The assumptions and the procedure of the derivation of the nonlinear turbulent electromotive force are as follows.
• We apply the multi-scale approach (Roberts & Soward 1975), which allows us to introduce fast and slow variables, and separate small-scale effects corresponding to fluctuations and large-scale effects describing mean fields.The mean fields depend on slow variables, while fluctuations depend on fast variables.Separation into slow and fast variables is widely used in theoretical physics, and all calculations are reduced to the Taylor expansions of all functions assuming that characteristic turbulent spatial and time scales are much smaller than the characteristic spatial and time scales of the mean magnetic field variations.
(2)-(3) written in a Fourier space, we derive equations for the second-order moments for the velocity field fij = uiuj , the magnetic field hij = bibj and the crosshelicity gij = uibj .
• We split the tensors fij , hij and gij into nonhelical hij and helical, h for magnetic fluctuations depends on the small-scale magnetic helicity, and its evolution is determined by the dynamic equation which follows from the magnetic helicity conservation arguments (Kleeorin & Ruzmaikin 1982;Gruzinov & Diamond 1994;Kleeorin et al. 1995;Kleeorin & Rogachevskii 1999;Kleeorin et al. 2000;Blackman & Brandenburg 2002).The characteristic time of the evolution of the nonhelical part of the magnetic tensor hij is of the order of the turbulent correlation time τ0 = ℓ0/u0, while the relaxation time of the helical part of the magnetic tensor h (H) ij is of the order of τ0 Rm, where Rm = ℓ0u0/η ≫ 1 is the magnetic Reynolds number, and u0 is the characteristic turbulent velocity in the integral scale ℓ0 of turbulent motions.
• Equations for the second-order moments contain higher-order moments and a problem of closing the equations for the higher-order moments arises.Various approximate methods have been proposed for the solution of this closure problem (Monin & Yaglom 1971, 2013;McComb 1990;Rogachevskii 2021).For small fluid and magnetic Reynolds numbers, the quasi-linear approach can be used (Rüdiger & Kichatinov 1993;Kitchatinov et al. 1994;Rüdiger et al. 2013), while for large fluid and magnetic Reynolds numbers, the minimal tau approach (Field et al. 1999) or the spectral τ approach (Rogachevskii & Kleeorin 2000, 2001, 2004, 2006) are applied to derive the nonlinear turbulent electromotive force.For instance, the spectral τ approach postulates that the deviations of the thirdorder moments, Mf (III) ij (k), from the contributions to these terms afforded by the background turbulence, Mf (III,0) ij (k), can be expressed through the similar deviations of the second-order moments, f (Orszag 1970;Pouquet et al. 1976;Kleeorin et al. 1990): where τr(k) is the scale-dependent relaxation time, which can be identified with the correlation time τ (k) of the turbulent velocity field for large fluid and magnetic Reynolds numbers.The superscript (0) corresponds to the background turbulence (with B = 0), and τr(k) is the characteristic relaxation time of the statistical moments.We apply the spectral τ approach only for the nonhelical part hij of the tensor for magnetic fluctuations.The spectral τ approach is widely used in the theory of kinetic equations, in passive scalar turbulence and magnetohydrodynamic turbulence.
• We use the following model for the second-order moment f (0) ij of isotropic inhomogeneous incompressible and helical background turbulence in a Fourier space: Here δij is the Kronecker tensor, kij = ki kj /k 2 and u • (∇× u) is the kinetic helicity.The energy spectrum function is in the inertial range of turbulence k0 k kν.Here the wave number k0 = 1/ℓ0, the length ℓ0 is the integral scale of turbulent motions, the wave number kν = ℓ −1 ν , the length ℓν = ℓ0Re −3/4 is the Kolmogorov (viscous) scale, and the expression for the turbulent correlation time is given by τ (k) = 2 τ0 (k/k0) −2/3 .The model for the second moment h (0) ij for magnetic fluctuations in a Fourier space caused by the small-scale dynamo (with a zero mean magnetic field) is We also take into account that the turbulent electromotive force is produced in a turbulence with a non-zero mean magnetic field, so that the cross-helicity tensor in the background turbulence vanishes, i.e., g ij = 0. • We assume that the characteristic time of variation of the mean magnetic field B is substantially larger than the correlation time τ (k) for all turbulence scales (which corresponds to the mean-field approach).This allows us to get a stationary solution for the equations for the second moments.Using the derived equations for the second moments fij , hij and gij, we determine the nonlinear turbulent electromotive force Ei = εimn gmn(k) dk.The details of the derivation of the nonlinear turbulent electromotive force are given by Rogachevskii & Kleeorin (2004).
For illustration of these results, we consider a smallscale homogeneous turbulence with a mean velocity shear, U = S z ey.We also consider, an axi-symmetric αΩ dynamo problem in the cartesian coordinates, so the mean magnetic field, B = By(x, z) ey + ∇×[A(x, z) ey], is determined by the following nonlinear dynamo equations (Rogachevskii & Kleeorin 2004): Here, the nonlinear α effect is given by where α (K) (B) is the kinetic α effect, and α (M) B is the magnetic α effect, which are given by Here α (0) ) characterised the small-scale dynamo is varied in the range 0 ǫ 1, where b 2 (0) /2µ0 and u 2 (0) /2 are turbulent magnetic and kinetic energies of the background turbulence, ℓ b is the characteristic scale of the localization of the magnetic energy due to the small-scale dynamo, and Hc B = b•(∇×b) is the current helicity of the small-scale magnetic field b.
The quenching functions φ K (β) and φ M (β) of the kinetic and magnetic α effects are given by Eqs.(A1)-(A2) in Appendix A.Here φ M (β) is the quenching function of the magnetic α effect derived by Field et al. (1999) using the minimal τ approximation (the τ approach applied in a physical space) and Rogachevskii & Kleeorin (2000) using the spectral τ approach.
The nonlinear turbulent magnetic diffusion coefficients for the poloidal η (A) T B and toroidal η (B) T B mean magnetic field are given by where η T = τ0 u 2 /3 is the characteristic value of the turbulent magnetic diffusivity.The quenching function φ and the functions φ K (β) and φ(β) are given by Eqs.(A1) and (A3) in Appendix A.Here for simplicity we consider a homogeneous background turbulence, so the effective pumping velocity of the large-scale magnetic field vanishes.
The asymptotic formulas for the kinetic and magnetic α effects, and the nonlinear turbulent magnetic diffusion coefficients of the mean magnetic field for a weak field B ≪ Beq/4 are given by and for a strong field B ≫ Beq/4 they are given by It follows from Eqs. ( 13)-( 19), that small-scale dynamo decreases the kinetic α effect and it increases the turbulent magnetic diffusion of the toroidal mean magnetic field.
As follows from Eq. ( 11), the magnetic α effect is proportional to the current helicity Hc B of the small-scale magnetic field (Pouquet et al. 1976), which describes the dynamical quenching of the α effect.Note that the dynamical quenching related to evolution of the magnetic α effect is derived only from the induction equation, and it is a contribution from small-scale current helicity b•(∇×b) , which is related to the small-scale magnetic helicity density.On the other hand, the algebraic quenching of the kinetic and magnetic alpha effects and turbulent magnetic diffusion coefficients of the large-scale magnetic field are derived from both, the Navier-Stokes equation for velocity fluctuations and the induction equation for magnetic fluctuations.In particular, the algebraic quenching is a contribution from the correlation functions for velocity fluctuations uiuj , magnetic fluctuations bibj and the cross-helicity correlation function uibj .The algebraic quenching is a physical effect related to a feedback of the growing large-scale magnetic field on plasma motions.If the algebraic quenching of the α effect is taken into account, the algebraic quenching of the turbulent magnetic diffusion should be taken into account as well.For instance, many studies related to the mean-field numerical simulations of the evolution of the solar and galactic magnetic fields take into account algebraic and dynamic quenching of the α effect, but ignore the algebraic quenching of the turbulent magnetic diffusion (see, e.g., Covas et al. 1997Covas et al. , 1998;;Kleeorin et al. 2000Kleeorin et al. , 2002Kleeorin et al. , 2003bKleeorin et al. , 2016Kleeorin et al. , 2020Kleeorin et al. , 2023 The approach discussed in this section allows us to derive the nonlinear turbulent electromotive force for an intermediate nonlinearity.This means that the mean magnetic field is not enough strong to affect the background turbulence.The theory for a strong mean magnetic field should take into account a modification of the background turbulence by the mean magnetic field. In the next sections we take into account this effect.In particular, we obtain the dependence of the turbulent kinetic energy ρ u 2 (0) /2 on the mean magnetic field using the budget equations for the turbulent kinetic and magnetic energies.This describes an additional nonlinear effect of the increase of the turbulent kinetic energy of the background turbulence by the dissipation of a strong mean magnetic field.The latter increases turbulent magnetic diffusion and decreases the nonlinear dynamo number for a strong field, resulting in a saturation of the dynamo growth of the largescale magnetic field.

BUDGET EQUATIONS
Using the Navier-Stokes equation for velocity fluctuations, we derive the budget equation for the density of turbulent kinetic energy (TKE), E K = ρ u 2 /2 as where is the dissipation rate of TKE, and is the production rate of TKE.Here U is the mean velocity, ν is the kinematic viscosity and the angular brackets imply ensemble averaging, F = s u is the turbulent flux of the entropy, s = θ/T + (γ −1 − 1)p/P are entropy fluctuations, θ and T are fluctuations and mean fluid temperature, ρ and ρ are fluctuations and mean fluid density, p and P are fluctuations and mean fluid pressure, γ = cp/cv is the ratio of specific heats, g is the acceleration due to the gravity and ρ f is the external steering force with a zero mean.We consider three different cases when turbulence is produced either by convection, or by large-scale shear motions or by an external steering force, see the last three terms in the RHS of Eq. ( 21).The first two terms in the RHS of Eq. ( 21) describe an energy exchange between the turbulent kinetic and magnetic energies (see below), and the third term in the RHS of Eq. ( 21) are due to the work of the Lorentz force in a nonuniform mean magnetic field.The estimate for the dissipation rate of the turbulent kinetic energy density in homogeneous isotropic and incompressible turbulence with a Kolmogorov spectrum is ε K = E K /τ0, where τ0 is the characteristic turbulent time at the integral scale.
Using the induction equation for magnetic fluctuations, we derive the budget equation for the density of turbulent magnetic energy (TME), E M = b 2 /2µ0 as where is the dissipation rate of TME, and is the production rate of TME.Here η is the magnetic diffusion due to electrical conductivity of the fluid.The first two terms in the RHS of Eq. ( 24) describe an energy exchange between the turbulent magnetic and kinetic energies.The estimate for the dissipation rate of the turbulent magnetic energy density is ε M = E M /τ0.The density of total turbulent energy (TTE), E T = E K + E M , is determined by the following budget equation: where is the production rate of E T , ε To determine the production rate of TTE, we use the following second moments for magnetic fluctuations (Rogachevskii & Kleeorin 2007) and velocity fluctuations, (see Appendix B), where βij = BiBj/B 2 .The tensor ui uj (0) for a background turbulence (with a zero mean magnetic field) in Eq. ( 28) has two contributions caused by background isotropic velocity fluctuations and tangling anisotropic velocity fluctuations due to the mean velocity shear (Elperin et al. 2002): where ∂U ij = (∇iU j + ∇jU i)/2 and ν (0) T = τ0 u 2 (0) /3 is the turbulent viscosity.For simplicity, in Eq. ( 27) we do not take into account a small-scale dynamo with a zero mean magnetic field.
The nonlinear functions qp(B) and qs(B) entering in Eq. ( 27)-( 28) are given by Eqs.(B6)-(B7) in Appendix B. The asymptotic formulae for the nonlinear functions qp(B) and qs(B) are as follows.For a very weak mean magnetic field, B ≪ Beq/4Rm and for B ≫ Beq/4 they are given by where β = √ 8 B/Beq.Substituting Eqs. ( 27)-( 29) into Eq.( 26), we obtain the production rate of the total turbulent energy as where E B = u × b is the turbulent nonlinear electromotive force.The turbulent viscosity ν T B depends on the mean magnetic field.In particular, for weak field B ≪ Beq/4, the turbulent viscosity ν T B ∼ ν (0) T = τ0 u 2 (0) /3, and for strong field B ≫ Beq/4, it is ν T B ∼ ν (0)  T (1+ǫ)/(4B/Beq) (Rogachevskii & Kleeorin 2007).Using the steady state solution of Eq. ( 25), we estimate the total turbulent energy density as E K + E M ∼ τ Π T , where τ is of the order of the turbulent time.Equation ( 27) yields the density of turbulent magnetic energy In the next sections, we apply the budget equations for analysis of nonlinear mean-field αΩ, α 2 and α 2 Ω dynamos.

MEAN-FIELD αΩ DYNAMO
In this section, we consider the axisymmetric mean-field αΩ dynamo, so that the mean magnetic field can be decomposed as and nonlinear mean-field induction equation reads where the operator N is given by and the operator describes differential rotation.Here ϑ is the angle between δΩ and the vertical coordinate z and L is the characteristic scale (e.g., the radius of a star or the thickness of a galactic disk).The total α effect is the sum of the kinetic α effect, α K (B), and the magnetic α effect, α M (B), , where the kinetic α effect is proportional to the kinetic helicity Hu = u•(∇×u) , and the magnetic α effect is proportional to the current helicity Hc B = b•(∇×b) of the small-scale magnetic field b.Equations ( 38)-( 40) are written in dimensionless variables: the coordinate is measured in the units of L, the time t is measured in the units of turbulent magnetic diffusion time L 2 /η (0)  T ; the mean magnetic field is measured in the units of B * , where and the magnetic potential, A is measured in the units of RαLB * .Here Rα = α * L/η (0) T , the fluid density ρ is measured in the units ρ * , the differential rotation δΩ is measured in units of the maximal value of the angular velocity Ω, the α effect is measured in units of the maximum value of the kinetic α effect, α * ; the integral scale of the turbulent motions ℓ0 = τ0 u0 and the characteristic turbulent velocity u0 = u 2 (0) at the scale ℓ0 are measured in units of their maximum values in the turbulent region, and the turbulent magnetic diffusion coefficients are measured in units of their maximum values.The magnetic Reynolds number Rm = ℓ0 u0/η is defined using the maximal values of the integral scale ℓ0 and the characteristic turbulent velocity u0.The dynamo number for the linear αΩ dynamo is defined as DL = RαRω, where Rω = (δΩ) L 2 /η (0) T .Now we define the nonlinear dynamo number DN B for the αΩ dynamo as where we take into account that the nonlinear turbulent magnetic diffusion coefficients of the poloidal and toroidal components of the mean magnetic field are different (Rogachevskii & Kleeorin 2004).
Next, we take into account the feedback of the mean magnetic field on the background turbulence using the budget equation for the total turbulent energy.In a shearproduced non-convective turbulence, the leading-order contributions to the production rate of the turbulent kinetic energy for a strong large-scale magnetic field (B ≫ Beq/4) is due to the term −E B •(∇×B)/µ0, so that the leadingorder contribution to the turbulent kinetic energy density for a strong large-scale magnetic field is estimated as Indeed, let us estimate the leading-order contributions to the production rate of the total turbulent energy given by ( 35).Using Eqs. ( 7)-( 8), we can rewrite the turbulent elec-tromotive force as Ei = αBi − η where (∇ × B)ϕ and (∇ × B)p are the toroidal and poloidal components of the electric current, which can be estimated as: Here the characteristic scale of the mean magnetic field variations LB is defined as LB = B/|∇×B|.We also take into account that for a strong field (B ≫ Beq/4), η (A) T /β, where Bϕ and Bp are the toroidal and poloidal components of the mean magnetic field.For the αΩ dynamo, the toroidal component of the mean magnetic field is much larger than the poloidal component, i.e., |Bp| ≪ |Bϕ|.This yields where the magnetic energy of the equipartition field Beq is defined as ] 1/2 being the integral scale of turbulence at vanishing mean magnetic field.We assume also that the correlation time is independent of the mean magnetic field.Contributions of other terms to the production rate of TTE and TKE for a strong large-scale magnetic field are much smaller than that described by Eq. ( 43).For instance, the contribution αB and for a strong field α(β) ∼ α (0) /β 2 .Similarly, the checking of the contributions of the remaining terms to the production rate of TTE and TKE for a strong large-scale magnetic field shows that they are much smaller than that described by Eq. ( 43).Therefore, the leading-order contribution to the turbulent kinetic energy density E K B for strong mean magnetic fields is Equation ( 44) implies that the turbulent kinetic energy increases due to the dissipation of the strong large-scale magnetic field.This yields the estimate for the turbulent magnetic diffusion coefficient of toroidal magnetic field η (B) where η (0) T = 2τ E (0) K /3ρ and we take into account the increase of the turbulent kinetic energy caused by the dissipation of the strong large-scale magnetic field [see Eq. ( 44)].As follows from Eq. ( 19), the ratio of turbulent magnetic diffusion coefficients of poloidal and toroidal fields The dependence of the total α effect on the mean magnetic field, α B , is caused by the algebraic and dynamic quenching.The algebraic quenching describes the feedback of the mean magnetic field on the plasma motions, while the dynamic quenching of the total α effect is caused by the evolution of the magnetic α effect related to the small-scale current and magnetic helicities.In particular, the dynamic equation for the small-scale current helicity (which determines the evolution of the magnetic α effect) in a steady state yields the estimate for the total α effect in the limit of a strong mean field as α B ∝ −divF M /B 2 , where F M is the magnetic helicity flux of the small-scale magnetic field.This implies that the total α effect for strong magnetic fields behaves as Note that the algebraic and dynamic quenching of the alpha effect yield similar behavior for a strong large-scale magnetic field [see Eqs. ( 17)-( 18) and ( 47), and paper by Chamandy et al. (2014)].Therefore, the ratio DN B /DL of the nonlinear and linear dynamo numbers in a shear-produced turbulence for strong mean magnetic fields is estimated as [see Eqs. ( 41) and ( 45)-( 47)]: Equation ( 48) implies that the nonlinear dynamo number decreases with the increase of the mean magnetic field for any strong values of the field for a shear-produced turbulence.This results in saturation of the mean-field dynamo instability.
In a convective turbulence, the largest contributions to the production rate of TTE for a strong mean magnetic fields is due to the buoyancy term ρ g Fz and the term η (B) T B (∇ × B) 2 /µ0 [see Eq. ( 35)].This implies that the leading-order contribution to the turbulent kinetic energy density E K B in a convective turbulence for strong mean magnetic fields is given by Eq. ( 44), where E (0) K = (ρ/2) (2g Fz ℓ0) 2/3 .Therefore, equations for the ratios η (B) T B and DN B /DL in a convective turbulence for strong mean magnetic fields are similar to Eqs. ( 45)-( 48), respectively.The difference is only in equation for E (0)   K that for a convective turbulence is given by E (0) K = (ρ/2) (2g Fz ℓ0) 2/3 and for a shear-produced turbulence is E (0) K = (2/3) ρ ℓ 2 0 S 2 .The similar situation is also for a forced turbulence except for the expression for E (0) K for a forced turbulence reads E (0) K = ρ τ0 u • f .This implies that for the αΩ dynamo, the nonlinear dynamo number decreases with increase of the mean magnetic field for a forced turbulence, and a shear-produced turbulence and a convective turbulence.This causes saturation of the mean-field αΩ dynamo instability for a strong mean magnetic field.

MEAN-FIELD α 2 DYNAMO
In this section, we consider mean-field α 2 dynamo.First, we discuss a long-standing question: "When can a onedimensional kinematic α 2 dynamo be oscillatory?"The mean magnetic field B(t, z) = ∇ × A = (−∇zAy, ∇zAx, 0) is determined by the following equation where A is the mean magnetic vector potential in the Weyl gauge.The linear operator L and the function Ψ(t, z) are given by where η (0) T is the turbulent magnetic diffusion coefficient, and α (0)  K is the kinetic α effect caused by the helical turbulent motions in plasma.If the linear operator L is not self-adjoint, it has complex eigenvalues.This case corresponds to the oscillatory growing solution, i.e., the dynamo is oscillatory.On the other hand, any self-adjoint operator, M , defining by the following condition, has real eigenvalues, where the asterisk denotes complex conjugation.Now we determine conditions when the linear operator L is not self-adjoint, i.e., it has complex eigenvalues.To this end, we determine the integrals Ψ * L Ψ dz and Ψ L * Ψ * dz as: where z = L bott and z = Ltop are the bottom and upper boundaries, respectively.When η (0) T and α (0) K vanish at the boundaries where the turbulence is very weak, the operator L satisfies condition (51) and the α 2 dynamo is not oscillatory.On the other hand, when α (0) K vanishes only at one boundary, while it is non-zero at the other boundary, the operator L does not satisfy condition (51), and the α 2 dynamo is oscillatory.The latter case has been considered in analytical study by Shukurov et al. (1985); Rädler & Bräuer (1987) and in numerical study by Baryshnikova & Shukurov (1987).Brandenburg (2017) has recently considered the onedimensional kinematic α 2 dynamo with different conditions at two boundaries: A = 0 at z = L bott and ∇zA = 0 at z = Ltop, so that the operator L may not satisfy condition (51), and the α 2 dynamo may be oscillatory.Now we consider the nonlinear axisymmetric mean-field α 2 dynamo, so that nonlinear mean-field induction equation reads where the mean magnetic field is B = By(t, x, z)ey + rot[A(t, x, z)ey], the operator N is given by and the total α effect is given by α B = α K B + α M B .Now we introduce the effective dynamo number D (α) N B in the nonlinear α 2 dynamo defined as Similarly, the effective dynamo number for a linear α 2 dynamo is defined as α , where Rα = α * L/η (0) T , α * is the maximum value of the kinetic α effect and L is the stellar radius or the thickness of the galactic disk.
The poloidal and toroidal components of the mean magnetic field in the nonlinear α 2 mean-field dynamo are of the same order of magnitude.Equations ( 44)-( 47) obtained in Section 4 can be used for the nonlinear α 2 mean-field dynamo as well.Therefore, the ratio D for strong mean magnetic fields is given by These equations take into account the feedback of the mean magnetic field on the background turbulence by means of the budget equation for the total turbulent energy.Thus, Eq. ( 56) implies that for the α 2 dynamo, the nonlinear dynamo number decreases with increase of the mean magnetic field.This causes a saturation of the mean-field α 2 dynamo instability for a strong mean magnetic field.
We consider a kinematic dynamo problem, assuming for simplicity that the kinetic α effect is a constant, and the mean velocity U = (0, Sz, 0), where S ≡ δΩ.We seek a solution for the linearised equation ( 57) as a real part of the following functions: where γ = γ +i ω.Equations ( 57)-( 60) yield the growth rate of the dynamo instability and the frequency of the dynamo waves as where Here we took into account that (x+iy) 1/2 = ±(X+iY ), where Here the threshold R cr α for the mean-field dynamo instability, defined by the conditions γ = 0 and Rω = 0, is given by R cr α = (k 2 x +k 2 z ) 1/2 .Equations ( 57)-( 60 In this case, the mean-field α 2 dynamo is slightly modified by a weak differential rotation, and the phase shift between the fields Bϕ and B pol vanishes, while B pol /Bϕ ∼ 1 [see Eqs. ( 64)-( 65)].In the opposite case, for a strong differential rotation, ζRω ≫ RαR cr α , the growth rate of the dynamo instability and the frequency of the dynamo waves are given by In this case, the mean-field αΩ dynamo is slightly modified by a weak α 2 effect, and the phase shift between the fields Bϕ and B pol tends to −π/4, while B pol /Bϕ ≪ 1 [see Eqs. ( 64)-( 65)].The necessary condition for the dynamo (γ > 0) in this case reads: • when Rα/R cr α < √ 2, the mean-field α 2 Ω dynamo is excited when • when Rα/R cr α > √ 2, the mean-field α 2 Ω dynamo is excited for any differential rotation, Rω.Here DL = Rα Rω.
Analysis which is similar to that performed in Section 4 [see Eqs. ( 44)-( 47)] yields the ratio of the nonlinear and linear dynamo numbers DN B /DL in the nonlinear α 2 Ω dynamo for strong mean magnetic fields that is coincided with Eq. ( 56).The latter implies that for the α 2 Ω dynamo, the nonlinear dynamo number decreases with increase of the mean magnetic field, so that the nonlinear mean-field dynamo instability is always saturated for strong mean magnetic fields.

CONCLUSIONS
In the sun, stars and galaxies, the large-scale magnetic fields are originated due to the mean-field dynamo instabilities.The saturation of the dynamo generated large-scale magnetic fields is caused by algebraic and dynamic nonlinearities.A key parameter which controls the saturation of the αΩ dynamo instability is the nonlinear dynamo number DN B = α B δΩ L 3 /η 2 T B .When the total α effect and the turbulent magnetic diffusion are quenched as (B/Beq) −2 for strong mean magnetic fields, the nonlinear dynamo number DN B increases with the increase of the large-scale magnetic field.The latter implies that the mean-field dynamo instability cannot be saturated for a strong field.
In the present study we have shown that the dissipation of the generated strong large-scale magnetic field increases both, the turbulent kinetic energy of the background turbulence and the turbulent magnetic diffusion coefficient.This additional nonlinear effect decreases the nonlinear dynamo number for a strong field and causes a saturation of the dynamo growth of large-scale magnetic field.This nonlinear effect is taken into account by means of the budget equation for the total turbulent energy.Using this approach and considering various origins of turbulence (e.g., a forced turbulence, a shear-produced turbulence and a convective turbulence), we have demonstrated that the mean-field αΩ, α 2 and α 2 Ω dynamo instabilities can be always saturated for any strong mean magnetic field.These results have very important applications for astrophysical magnetic fields.

ACKNOWLEDGMENTS
The detailed comments on our manuscript by the anonymous referee which essentially improved the presentation of ) also yield the squared ratio of amplitudes |A0/B0| 2 , shift δ between the toroidal field Bϕ and the magnetic vector potential A is given by sin(2δ) = −ζRω (RαR cr α ) 2 + ζ 2 R yields the energy ratio of poloidal B pol and toroidal Bϕ mean magnetic field components as B cr α A) 2 .Asymptotic formulas for the growth rate of the dynamo instability and the frequency of the dynamo waves for a weak differential rotation, ζRω ≪ RαR cr α , are given by γ = RαR cr α