Stability problem in dynamo

It is shown, that the saturated $\alpha$-effect taken from the nonlinear dynamo equations for the thin disk can still produce exponentially growing magnetic field in the case, when this field does not feed back on the $\alpha$. For negative dynamo number (stationary regime) stability is defined by the structure of the spectra of the linear problem for the positive dynamo numbers. Stability condition for the oscillatory solution (positive dynamo number) is also obtained and related to the phase shift of the original magnetic field, which produced saturated $\alpha$ and magnetic field in the kinematic regime. Results can be used for explanation of the similar effect observed in the shell models simulations as well in the 3D dynamo models in the plane layer and sphere.


Introduction
It is believed, that variety of the magnetic fields observed in astrophysics and technics can be explained in terms of the dynamo theory, e.g. (Hollerbach & Rüdiger, 2004). The main idea is that kinetic energy of the conductive motions is transformed into the energy of the magnetic field. Magnetic field generation is the threshold phenomenon: it starts when magnetic Reynolds number R m reaches its critical value R cr m . After that magnetic field grows exponentially up to the moment, when it already can feed back on the flow. This influence does not come to the simple suppression of the motions and reducing of R m , rather to the change of the spectra of the fields closely connected to constraints caused by conservation of the magnetic energy and helicity (Brandenburg & Subramanian, 2005). The other important point is effects of the phase shift and coherence of the physical fields before and after onset of quenching discussed in (Tilgner & Brandenburg, 2008).
As a result, even after quenching the saturated velocity field is still large enough, so that R m ≫ R cr m . Moreover, velocity field taken from the nonlinear problem (when the exponential growth of the magnetic field stopped) can still generate exponentially growing magnetic field providing that feed back of the magnetic field on the flow is omitted (kinematic dynamo regime) (Cattaneo & Tobias, 2009;Tilgner, 2008;Tilgner & Brandenburg, 2008;Schrinner, Schmidt, Cameron, 2009). In other words, the problem of stability of the full dynamo equations including induction equation, the Navier-Stokes equation with the Lorentz force differs from the stability problem of the single induction equation with the given saturated velocity field taken from the full dynamo solution: stability of the first problem does not provide stability of the second one.
Here we consider effect of such kind of stability on an example of the model of galactic dynamo in the thin disk, as well as some applications to the dynamo in the sphere.

Dynamo in the thin disk
One of the simplest galactic dynamo models is a one-dimensional model in the thin disk (Ruzmaikin, Shukurov, Sokoloff, 1988): where A and B are azimuthal components of the vector potential and magnetic field, α(z) is a kinetic helicity, D is a dynamo number, which is a product of the amplitudes of the αand ω-effects and primes denote derivatives with respect to a cylindrical polar coordinate z. Equation (1) is solved in the interval −1 ≤ z ≤ 1 with the boundary conditions B = 0 and A ′ = 0 at z = ±1. We look for a solution of the form Substituting (2) in (1) yields the following eigenvalue problem: where the constant γ is the growth rate. So as α(−z) = −α(z) is odd function of z, the generation equations have an important property: system (3) is invariant under transformation z → −z when (Parker, 1971): Therefore, all solutions may be divided into two groups: odd on B(z), dipole (D), and even, quadrupole on B(z).
System (3) has growing solution, ℜγ > 0, when |D| > |D cr |. For D < 0 the first exciting mode is quadrupole with D cr ≈ −8 and ℑγ = 0: solution is non-oscillatory 1 . For D > 0 the leading mode is oscillatory dipole, ℑγ 0 with higher threshold of generation: D cr ∼ 200. Putting nonlinearity of the form in (1), where E m = (B 2 + A ′2 )/2 is a magnetic energy, gives stationary solutions for Q-kind of symmetry and quasi-stationary solutions for D, see about various forms of nonlinearities in (Beck, Brandenburg, Moss, Shukurov, Sokoloff, 1996). The property of the nonlinear solution is mostly defined by the form of the first eigenfunction. Now, in the spirit of (Cattaneo & Tobias, 2009;Tilgner & Brandenburg, 2008) we add to (1) equations for the new magnetic field ( A, B) with the same α (5), which For D > 0 situation is different, resembling that one of instability described in (Cattaneo & Tobias, 2009;Tilgner, 2008;Tilgner & Brandenburg, 2008;Schrinner, Schmidt, Cameron, 2009) for more sophisticated models: field ( A, B) oscillates and starts to grow exponentially, see Fig. 2. Note, that no regime in oscillations for ( A, B) is observed. The other specific feature is delay of ( (5) is averaged over the space, so that α is steady, then instability dissapears. The question arises: does instability depends on stationarity, either it depends on something else?
It is known, that for D < 0 stability of the system (3, 5), which has stationary solution, is tightly bound to behaviour of the linear solution of (3) for D > 0 (Reshetnyak, Sokoloff, Shukurov, 1992). Note, that for the complex form of (3) it is equivalent to the solution of the conjugate problem.
where α e = α + ∂α ∂B B for α = α 0 1 + B 2 is Behaviour of αω-dynamo (3) is defined by the sign of Dα, and its change in the perturbed equations (7) A, B). Note, that D cr + ≪ D e does not guarantee, that ( A, B) will grow exponentially due to nonlinearity (5).
It is worthy of note that nonlinear solution of (1,5) and (6,5) demonstrates similar stationary behaviour even for D ∼ −10 3 in spite of the fact, that D cr + for the quadrupole oscillatory mode for positive D is ∼ 200. The reason is that dynamo system tends to the state of the strong magnetic filed with B ∼ D 1/2 , so that α ∼ 1 B 2 , leaving D e at the level of the first mode's threshold of generation.
For positive D (A, B), and therefore α(B), oscillate and one needs additional information on correlation of the waves. Here, instead of (8) we get α e ∼ − α 0 B |B| 3 . If phase shift between B and B is negligible, then α-effect is saturated and time evolution of (A, B) and ( A, B) is similar. However, simulations demonstrate Fig. 2, that field ( A, B) delays relative to (A, B). This is typical situation, when parameter resonance takes place: α is modulated by signal with frequency Ω ∼ 2ω, ω = ℑγ, see (Dawes & Proctor, 2008) for details of spatial resonance. This assumption is supported by the fact, that instability disappears when in (5) steady α, averaged in time, is used. Note, that usage of quadrupole boundary conditions in (6) is not important for instability: problem (6,5) with periodical boundary conditions has oscillatory solution and instability depends on the form of quenching in the same way.
To demonstrate what happens we consider how delay θ of ( A, B) relative to (A, B) changes production of A 2 + B 2 near the threshold of generation D cr + . We start from the linear analysis of the system in the form: From condition of solvability for (9): (k 2 + iω) 2 = −iD cr + kα 0 with α = α 0 follows that ω 2 = k 4 = 1. The other prediction of the linear analysis is the phase shift ϕ between A and B: ϕ = ± π 4 , what is twice smaller than for the nonlinear regime (Tilgner & Brandenburg, 2008), so that for the nonlinear regime the maximal A is when B is zero and quenching is absent.

Conclusions
Here we argue, that stability of the kinematic αω-dynamo problem with the α-effect taken from the the weaklynonlinear regime near the threshold of generation can be predicted from the knowledge on the threshold of generation of the linear problem with the opposite sign of the dynamo number. It appears, that in spite of the fact, that the magnetic field already saturated α, it still can generate magnetic field if spectra of linear problem are similar for dynamo number D with the opposite sign. So, as D depends on the product of the α and ω effects similar analysis can be performed with the ω-quenching, usually used in geodynamo models, see e.g. (Soward, 1978), as well as with the feed back of the magnetic field on diffusion. It is likely, that for the more complex systems, velocity field, taken from the saturated regime, with many exited modes will always generate magnetic field if the Lorentz force would be omitted.
So as nonlinearity (5) has quite a general form, we consider applications of these results to some other dynamo models.
Linear analysis of the axi-symmetrical αω-equations gives the following, see (Moffatt, 1978) and references therein: for positive D (which is believed to be in the Earth) in presence of the meridional velocity U p the first exciting mode is dipole with ℑγ = 0. Reducing of U p firstly leads to oscillatory dipole solution (regime of the frequent reversals (Braginskii, 1964)). The further reduce of U p gives the quadrupole oscillatory regime with larger value of D cr . For negative D and U p 0 the first mode is quadrupole with ℑγ = 0. U p → 0 gives non-oscillatory dipole mode with decreased D cr , see for more details (Meunier, Proctor, Sokoloff, Soward, Tobias, 1997). In contrast to the dynamo in the disk the thresholds of generation for positive and negative D in the sphere are of the same order and situation with stability of the field B is uncertain, and can depend on the particular form of the αand ω-effects. Anyway, stability of B for the steady regime is more expected.
In accordance with (Cattaneo & Tobias, 2009) shell models of turbulence demonstrate exponential growth of the magnetic field. This case, as well as 3D simulations of the turbulence in the box, which have the same instabilities, correspond to the oscillatory regimes and using our predictions should be unstable.
In the case of the 3D dynamo in the sphere simulations demonstrate different behaviour of B (Tilgner, 2008;Schrinner, Schmidt, Cameron, 2009). For small Rayleigh numbers, when the preferred solutions is dipole (in oscillations) and close to the single mode structure (Case 1 in (Christensen, Aubert, Cardin, Dormy, Gibbons et al., 2001)) B is finite. Increase of the Rossby number (Schrinner, Schmidt, Cameron, 2009) leads to the turbulent state and B becomes unstable.
Author is grateful to A.Brandenburg and D.Sokoloff for discussions.