Mean-field effects in the Galloway-Proctor flow

The coefficients defining the mean electromotive force in a Galloway-Proctor flow are determined. This flow shows a two-dimensional pattern and is helical. The pattern wobbles in its plane. Apart from one exception a circular motion of the flow pattern is assumed. This corresponds to one of the cases considered recently by Courvoisier, Hughes and Tobias (2006, Phys. Rev. Lett., 96, 034503). An analytic theory of the alpha effect and related effects in this flow is developed within the second-order correlation approximation and a corresponding fourth-order approximation. In the validity range of these approximations there is an alpha effect but no gamma effect, or pumping effect. Numerical results obtained with the test-field method, which are independent of these approximations, confirm the results for alpha and show that gamma is in general nonzero. Both alpha and gamma show a complex dependency on the magnetic Reynolds number and other parameters that define the flow, that is, amplitude and frequency of the wobbling motion. Some results for the magnetic diffusivity eta_t and a related quantity are given, too. Finally a result for alpha in the case of a randomly varying flow without the aforementioned circular motion is presented. This flow may be a more appropriate model for studying the alpha effect and related effects in flows that are statistical isotropic in a plane.


INTRODUCTION
In the astrophysical context, turbulent flows, e.g. in stellar convection zones or in accretion discs and galaxies, are generally anisotropic and time-dependent. A simple model of a flow with such properties is that by Galloway & Proctor (1992). This flow is two-dimensional, depends only on two Cartesian coordinates, e.g. x and y, which can simplify the analysis significantly, even in dynamo problems that are inherently three-dimensional. The Galloway-Proctor (GP) flow is related to a flow considered by Roberts (1972). The Roberts flow 1 is an early example of a spatially periodic flow that produces an alpha effect. The alpha term in the averaged form of the induction equation is crucial to model the generation of large-scale magnetic fields from small-scale helical fluid motions in stars and galaxies; see, for example, Moffatt (1978), Parker (1979), and Krause & Rädler (1980) for standard references. However, unlike the Roberts flow, the GP flow is timedependent with a flow pattern wobbling in the (x, y) plane in a circular fashion. Both the GP flow and the Roberts flow have a velocity component out of this plane such that the flow can be fully helical, i.e. the velocity is proportional to its curl.
Particularly important is the dependence of the α effect on the magnetic Reynolds number, Rm. While for the Roberts flow α declines with Rm in the large Rm limit, in the case of the GP flow ⋆ E-mail: khraedler@arcor.de (KHR); brandenb@nordita.org (AB) 1 As usual, the term Roberts flow refers to the flow given by equation (5.1) of Roberts (1972). according to the results by Courvoisier, Hughes & Tobias (2006) (in the following referred to as CHT06) and Courvoisier (2008) there is a more complicated dependence on Rm with sign changes and no indication of convergence with increasing Rm.
In many studies turbulent astrophysical flows have been modelled by random forcing. In the case of of helical isotropic turbulence such investigations show that α approaches a finite value as soon as Rm exceeds a value of the order of unity. This has been observed at least for Reynolds numbers up to 200 (Sur et al. 2008).
The purpose of this paper is to study the effects of the GP flow in more detail in order to understand the influence of timedependence and anisotropy on the value of α and other turbulent transport coefficients. In particular, it is important to document the differences and similarities with turbulent flows that are statistically isotropic and irregular in space and time. We focus attention here on the simplest case considered in CHT06 with a flow being purely periodic in time and add a few results for a simple flow with random time dependence.
A number of similarities, but also some striking differences between turbulent flows and the Roberts flow are known. Similar in both flows is the fact that there is an α effect whose magnitude increases with Rm as long as the latter does not exceed some value in the order of unity. However, for larger Rm, the α coefficient in isotropic turbulence settles to a constant value (Sur et al. 2008), while for the Roberts flow α tends to zero as Rm → ∞ (Soward 1987(Soward , 1989Rädler et al. 2002a,b). Furthermore, there is no γ effect, or pumping effect, neither for isotropic turbulence nor for the Roberts flow. On the other hand, in the time-dependent GP flow γ effects have been reported (CHT06). A γ effect corresponds to antisymmetric contributions of the α tensor. This raises the question about the possible existence of antisymmetric contributions to the turbulent magnetic diffusivity tensor, or ηt tensor.
These aspects are now straightforward to address using the recently developed test-field method to calculate numerically all components of the α and ηt tensors defining the mean electromotive force E for a given flow field (Schrinner et al. 2005(Schrinner et al. , 2007. If, as we assume here, too, the mean magnetic field B depends only on one of the Cartesian coordinates, say z, only two 2 × 2 tensors for α and ηt are of interest. The test-field method has recently been used to calculate diagonal and off-diagonal components of ηt , the magnetic Reynolds number dependence of α and ηt (Sur et al. 2008), as well as their scale dependence (Brandenburg, Rädler & Schrinner 2008). We begin by exploring general properties of the mean electromotive force in the GP flow and present analytical results for coefficients like α and γ, which are crucial for the electromotive force, gained in the second-order correlation approximation and in a corresponding fourth-order approximation. After explaining the test-field method we give a series of numerical results for such coefficients, which are independent of approximations of that kind, and discuss them in detail.

Definition of the problem
Consider a magnetic field B in an infinitely extended homogeneous conducting fluid with constant magnetic diffusivity η moving with a velocity u. Its behavior is governed by Referring to a Cartesian coordinate system (x, y, z) the velocity u is specified by with ψ = u0 kH cos(kHx + ϕx) + cos(kHy + ϕy) .
Hereẑ means the unit vector in the z direction, kH is a positive constant such that 2π/kH is the length of the diagonal of a flow cell, and ϕx and ϕy are functions of time to be specified later. Further we have u0 = urms/ √ 2. In the special case ϕx = ϕy = 0 the flow agrees with a Roberts flow. For non-zero ϕx or ϕy a properly moving frame of reference can be found in which we have again a steady Roberts flow pattern. In our original frame each point of this pattern moves with the velocity −k −1 H (∂tϕx, ∂tϕy) in the xy plane. In (2) the ratio of the flow components in the xy plane and in z direction has been fixed such that the modulus of the average of the kinetic helicity u · (∇ × u) over all x and y for given u0 takes its maximum. With the signs chosen this average is equal to −2u 2 0 kH. In view of the first example treated in CHT06 we specify the flow generally defined by (2) and (3) further to be a Galloway-Proctor flow and put ϕx = ǫ cos ωt , ϕy = ǫ sin ωt , where ǫ and ω are considered as non-negative constants. We label this flow in what follows by (i). Each point of this pattern moves with the frequency ω/2π on a circle with the radius ǫ/kH. To come closer to a turbulent situation CHT06 added a random function of time to the arguments ωt in (4). Another case of some interest occurs if we simply interpret ϕx and ϕy as random functions. More precisely we put ϕx = ǫφx(t/τc) , ϕy = ǫφy(t/τc) , where ǫ is again a constant, φx and φy are two independent but statistically equivalent random functions, which take positive and negative values between −1 and 1 and tend to zero with growing moduli of the argument, and τc is some correlation time. We label this random flow by (ii).

Mean-field concept
Adopting the mean-field concept, we denote mean fields by an overbar and define them as averages over all x and y. We have then u = 0. Taking the average of (1) we find with the mean electromotive force where [In (6) ∇ reduces simply to (0, 0, ∂z).] From (1) and (6) we conclude that b has to obey We adopt here the assumption that the mean electromotive force E is, apart from u and η, completely determined by B and its first spatial derivatives. (This assumption will be relaxed in Sect. 4.) This implies that there is no small-scale dynamo and that sufficient time has elapsed since the initial instant so that E no longer depends on any initial conditions. Since B is by definition independent of x and y its spatial derivatives can be represented by ∇ × B. We write simply J instead of ∇ × B, being aware that the mean electric current density is really ∇ × B/µ (rather than J ), where µ is magnetic permeability of the conducting fluid. Clearly we have now J = (−∂By/∂z, ∂Bx/∂z, 0). For the sake of simplicity we further assume that B is steady. In the so defined framework we may write with tensors αij and ηij determined by u and η only. Both αij and ηij , and so Ei, too, depend in general on time. We see from (8) that, if B is a uniform field, b and therefore E are independent of Bz. Hence we have αi3 = 0. Furthermore, since Jz = 0, clearly ηi3 is without interest, and we put ηi3 = 0.

Mean electromotive force in case (i)
For a more detailed investigation of E we focus on the fluid flow of type (i). In this case αij and ηij are periodic in time with a basic period equal to that of u, that is 2π/ω, or (as we will see below) a fraction of it.
Remarkably the velocity field u = u(x, y, t) defined by (2), (3) and (4) is invariant under a 90 • rotation about the z axis and a simultaneous retarding by π/2ω (that is, ωt → ωt − π/2). Consequently the αij in the correspondingly rotated coordinate system, which we denote by α ′ ij , have to satisfy the relation If we consider for a moment the change of the spatial coordinate system only and ignore any time dependence we have α ′ 11 = α22, α ′ 12 = −α21, α ′ 21 = −α12, α ′ 22 = α11, α ′ 31 = −α32 and α ′ 32 = α31. Hence (10) provides us with From the first two lines we conclude firstly that α11, α12, α21 and α22 have as functions of time a basic period of π/ω (not 2π/ω) and that α22(t) = α11(t ± π/2ω) and α21(t) = −α12(t ± π/2ω). The last line of (11) tells us that the averages of α31 and α32 over the period 2π/ω vanish so that α31 and α32 are simply oscillations around zero, and that they change their signs under time shifts by π/ω. Our reasoning for αij applies analogously to ηij .
We write down the result of these considerations in the form Hereα,γ,ηt andδ are in general periodic functions of time with the basic period π/ω, butκ andλ are periodic functions with period 2π/ω, which show sign changes under any time shift by π/ω and vanish under averaging over the period 2π/ω. From (9) and (12) we conclude Hereα † ,γ † ,η † t andδ † differ only by a phase shift of π/2 fromα, γ,ηt andδ, respectively, andκ † andλ † by a phase shift of π from κ andλ.
In addition to fields as defined above by averaging over x and y we consider also time-averaged mean fields defined by additional averaging over a time interval of length 2π/ω (but we refer to them only if explicitly indicated). When speaking of time averaging in what follows we always refer to this interval. For time-averaged mean fields (13) turns into where α, γ, ηt and δ are time averages ofα,γ,ηt andδ. 2 In the special case of the Roberts flow, i.e. ǫ = 0, the coefficientα is independent of time and so coincides withα † , and this applies analogously toγ,η,δ,κ andλ. In addition in this case the inversion ofẑ in (2) is equivalent to a shift of the flow pattern, e.g., by π/ √ 2 kH along y = x. Since such a shift does not change averages, E as given by (14) must be even inẑ. Therefore we have then γ = δ = 0. Nonzero γ and δ terms in (14) require a break of this 2 In view of the signs of α and γ we deviate here from representations as given, e.g., in Rädler et al. (2002a) but follow CHT06. symmetry, that is, a preference ofẑ over −ẑ, and this may occur as a consequence of the aforementioned circular motion of the flow pattern.
We override for a moment our restriction to non-negative values of the frequency ω and admit also negative ones. For ω > 0 the circular motion of the flow pattern defines, together with the z direction, a right-handed screw, and for ω < 0 a left-handed one. We conclude from this fact that inversion of the sign of ω has no other consequences than inversion of the signs of γ and δ.

Second-order approximation
The task of determination of E is now reduced to the determination of the six functionsα,γ,ηt,δ,κ andλ which occur in (12) and (13). As a first step in that direction we investigate E within the second-order correlation approximation (SOCA). Later we will proceed to a corresponding fourth-order approximation.
SOCA is defined by the neglect of the term with u×b−u × b on the right-hand side of equation (8) for b, which turns so into We may solve this equation with u as given by (2) and (3) analytically and calculate then E , see Appendix B. When choosing the form (13) of the result we havẽ γ =δ =κ =λ = 0 .
Here we have used the definition and χ (2) is given by and The parameter q gives, apart from a factor 2π, the ratio of the decay time of a magnetic structure with a length scale 2π/kH, that is (2π) 2 /ηk 2 H , and the wobble period 2π/ω of the flow pattern. In the case of small q the magnetic field follows the fluid motion immediately, but for large q it does so only with large delay. These two cases are sometimes labelled as "low conductivity limit" and "high conductivity limit", respectively.
In agreement with the general findings summarized in (12), the function χ (2) is periodic in time with a basic period π/ω. Whereasα andα † differ by a phase shift of π/2,ηt andη † t coincide. χ (2) satisfies |χ (2) | ≤ 1. It must be positive as long as ǫ ≤ π/4 but may otherwise take negative values, too. If ǫ = 0, or ǫ = 0 and q = 0 (what corresponds to the low-conductivity limit), χ (2) is independent of time and equal to unity. In Appendix B some numerically determined values of χ (2) are given. We note further that and For time-averaged mean fields we have again (14), now with where χ (2) 0 means the time average of χ (2) over the period π/ω. We point out that the time average of a function of ωt, say f (ωt), over an interval of the length π/ω is independent of ω. This is obvious from (ω/π) does not explicitly, but only in a indirect way via q, depend on ω.
As we know from general considerations on SOCA (e.g., Krause & Rädler 1980) the range of applicability of SOCA depends on q. For small q a sufficient condition for its validity reads Rm ≪ 1. For large q such a condition is Rm/q ≪ 1.

Higher-order approximations
Going now beyond SOCA we start again with Eq. (8) for b and put with b (n) being of the order n in u, and correspondingly In that sense b and E in Sect. 2.3 have to be interpreted as b (1) and E (2) , respectively. From (8) and (26) we obtain (15), now with b (1) instead of b, and further Using our result for b (1) and (28) In the same way we may calculate b (3) and E (4) . However, these calculations are rather tedious. For the sake of simplicity we have ignored all contributions to E (4) resulting from derivatives of B, that is, the terms withηt,δ andλ in (13). Some details of the calculations are explained in Appendix C.
Considering the results of all approximations up to the fourth order and referring again to (13) we have now The functions χ (4 α) and χ (4 γ) are given by with CC as defined by (19) and analogously defined quantities CS, SC and SS, Note that CC and CS are symmetric but SC and SS antisymmetric in the two arguments. Like χ (2) both χ (4α) and χ (4γ) oscillate with a basic period π/ω. They satisfy |χ (4α) | ≤ 1 and |χ (4γ) | ≤ 1. Further χ (4α) is positive as long as |ǫ| < π/4. In contrast to χ (4α) , however, the time average of χ (4γ) over a period π/ω is equal to zero. Whereas χ (4α) is even, χ (4γ) is odd in ω. We have further and For time-averaged mean fields again relation (14) applies, now with where does not explicitly depend on ω. Unfortunately, values for ηt and δ are not available. We have and With (34) we find then Results of higher approximations are very desirable but require heavy efforts. We suspect that in the approximation of sixth order in u the time averages ofγ andδ, and so the coefficients γ and δ in (14) no longer vanishes. This presumption is supported by numerical results (see below).

Mean electromotive force in case (ii)
Modifying the considerations on case (i) properly we may conclude that relation (14), again considered for time-averaged fields, applies for the fluid flow of type (ii) with γ = δ = 0. By contrast to case (i) the correlation between velocity components at different times vanishes if the time difference becomes very large.
Modifying also the SOCA calculations described above and in Appendix B correspondingly we find again (23), but with χ (2) 0 being the time average of (2) 0 on q for two different ǫ, calculated numerically on the basis of equation (38). The dashed line shows that χ where q is now defined by We have here again χ (2) = 1 for q = 0, and χ (2) vanishes for q → ∞. Like χ (2) also χ (2) 0 depends on ǫ and q but no longer explicitly on τc. We have calculated χ (2) 0 on the basis of equation (38) under the assumption that φx is always constant over time intervals of a given length. Fig. 1 shows dependencies on ǫ and q.

TEST-FIELD METHOD
We will determine numerically the elements of the tensors αij and ηij introduced with (9), but with 1 ≤ i, j ≤ 2 only, employing the test-field method of Schrinner et al. (2005Schrinner et al. ( , 2007. We will calculate E = u × b from numerical solutions b of (8), with B replaced by one out of four test fields B pq , where B and k are a constants. Repeating this for all test fields, denoting the E that belongs to a given B pq by E pq , and using (9) we find From this we conclude again for 1 ≤ i, j ≤ 2.
We point out that, although the E pq depend on z, the αij and ηij have to be independent of z. We further note that relation (9), on which these considerations are based, can only be justified under the assumption that all higher than first-order spatial derivatives of B are negligible. The derivatives of order n of our test fields B pq are proportional to k n . For this reason the results (43) apply in a strict sense only in the limit k → 0 (cf. Brandenburg, Rädler & Schrinner 2008). Let us focus here on case (i). After having calculated the αij and ηij in the way indicated above we may determine theα,γ,ηt andδ according to (12), that is, We are, however, mainly interested in the time-independent coefficients α, γ, η and δ that are relevant for time-averaged mean fields as addressed in (14). They are just time averages of theα,γ,ηt and δ, that is where · · · means averaging over a time interval of length π/ω. In case (ii) the relations (45) apply with · · · interpreted as averaging over a sufficiently long time.

A GENERALIZATION
So far we have assumed that the mean electromotive force E in a given point is completely determined by B and its first spatial derivatives in this point. If we relax this assumption we may proceed as in Brandenburg, Rädler & Schrinner (2008). In that sense we may replace (9), applied to time-averaged mean fields, by with kernelsαij andηij . When using a Fourier transformation Q(z) = Q exp(ikz)dz, this turns intõ wherẽ αij (k) = αij (ζ) cos kζ dζ ,ηij (k) = ηij (ζ) cos kζ dζ . (48) In this understanding the relations (41)-(43) apply with αij and ηij being replaced byαij andηij , which have a well-defined meaning for all k (not only in the limit k → ∞).

Units and dimensionless parameters
It is appropriate to giveα andγ as well as α and γ in units of u0, andηt,δ, ηt and δ in units of u0/kH. The remaining dimensionless parts of these coefficients are then, apart from the time dependencies ofα,γ,ηt andδ, functions of the dimensionless parameters Rm, ǫ and q introduced through (17), (3), and either (20) or (39). Instead of q we may also use the dimensionless quantityω defined bỹ Figure 2. Time dependence of α ij for the parameters used in Fig. 1 of CHT06 which, in our normalization, are Rm = 78, ǫ = 3/4, andω = 2/3. Here, ∆t = t − t 0 , where t 0 = 300/ω is the final time shown in Fig. 1 of CHT06. The dotted lines refer to α 11 and α 21 , respectively, the dashed lines to α 12 and α 22 , and the dash-dotted lines to (α 11 + α 22 )/2 and (α 21 − α 12 )/2. The straight solid lines give the time averages of the latter quantities, that is, α and γ.
In case (i) we have soω = ω/u0kH, which is the ratio of the turnover time (u0kH/2π) −1 to the wobble period 2π/ω. In case (ii) appliesω = (τcu0kH) −1 , and this is, apart from a factor 2π, the ratio of that turnover time to the time τc introduced with the random flow.

Comparison with CHT06
We show first that our method reproduces results by CHT06. We suppose that our Rm is related to the magnetic Reynolds number, say R CHT m , used but not explicitly defined there, by Rm = 3/2 R CHT m . While in CHT06 dependencies of the results on Rm and ǫ are considered, no values of q orω are given. We suppose that the calculations have actually been carried out withω = 2/3. Finally we suppose that the unit of α and γ used by CHT06 is ω/kH.
With a view to Fig. 1 of CHT06 we have carried out calculations with Rm = 3/2 × 64 ≈ 78, ǫ = 3/4 andω = 2/3. Our results for the αij obtained with these parameters and given in this particular case in units of ω/kH are presented in our Fig. 2. We see in particular that α11 and α22 vary between −6 and −1 with a period π/ω. As far as α11 is concerned this agrees with the result for u × b x shown in Fig. 1 of CHT06. Also the initial evolution of α11, which is not shown here, agrees with this figure. Furthermore, in our Fig. 2 the phase shift by π/2 between α11 and α22 discussed in Sect. 2.2 is clearly visible. Our results for α12 and α21 lead to a value of γ, which agrees in modulus but differs in sign from that of CHT06. (To obtain their sign we need to replace ω by −ω.) With the above values of Rm andω but ǫ = 1 we find again a sign of γ opposite to that of CHT06.  (37), which has been derived for ǫq ≪ 1 and q ≪ 1 only.

Time-averaged mean fields
Switching now to time-averaged mean fields we start with Fig. 3, which shows results for α at Rm = 0.1 in dependence on ǫq. They were found with the help of numerical integrations of the test-field version of (8) in its complete form or after reducing it to SOCA. It turned out that SOCA is sufficient for their calculation. Some of these results were also confirmed by evaluating (23) with (18) or (24). As long as ǫq is small, α depends in agreement with (23) and (24) only via this product on ǫ and q. For larger ǫq it depends, however, in a more complex way on ǫ and q. Furthermore, α remains finite if ǫ = 1 and q grows, and it tends to zero if q = 0.1 and ǫ grows. Since Rm is small the validity of SOCA is plausible in the case q = 0.1. It is however remarkable in that with ǫ = 1, in which q may grow up to 10.
Next, we consider the dependence of α and γ on Rm, in Fig. 4 shown for ǫ = 1 andω = 1 (i.e. q = Rm). For small Rm we expect that SOCA applies and so α/u0 is linear in Rm but γ vanishes. Indeed α/u0 shows this linearity up to Rm ≈ 1. In agreement with the results of CHT06 γ is negative and its modulus remains small for Rm < 1. Remarkably the values of α/u0 calculated from (23) and (24) (dotted line), or (37) (dashed line), which have been derived for q ≪ 1 and ǫq ≪ 1, deviate for Rm > 1 drastically from both the numerically obtained SOCA results (dash-dotted line) and those obtained without any approximation of that kind (solid line). The proportionality of γ/u0 with R 5 m confirms the presumption made at the end of Sect. 2.3 that nonzero values of γ occur only in sixth-order and higher approximations with respect to u0.
Simple arguments (as given in Sect. 6 below) suggest that α is never negative. However, CHT06 found that not only the moduli but also the signs of both α and γ depend for each given Rm sensitively on ǫ. In our Fig. 5, which applies for Rm = 100 and ω = 2/3, both α and γ vary strongly with ǫ, too. The represented results confirm, apart from the sign of γ, the corresponding ones in Fig. 2 of CHT06. Both α and γ change their signs with ǫ. As Fig. 6 shows, in the situation with the same Rm andω = 1 only γ changes its sign, which indicates a considerable effect of changingω. In both of the cases considered in Fig. 5 and Fig. 6, α and γ diminish for small as well as large values of ǫ.
In Fig. 7 and Fig. 8 we see that α and γ depend, at least for Rm = 100 and ǫ = 1, also sensitively on the parameterω, or q, that is, on the frequency with which the velocity pattern wobbles.  (23) and (24), or (37), respectively.    There are, however, simple asymptotic behaviors for small and for largeω, clearly visible forω < 0.1 andω > 3. Similar results have been found for Rm = 10 and ǫ = 1. In this case, however, α stays positive for all values ofω, and only one sign reversal of γ occurs. We see from CHT06 that there is a rich dependence of α and γ on Rm for values ofω and ǫ of order unity. In Fig. 9 we show results for an example withω = 0.5. Reversals of α are then possible for rather small values of Rm of the order of 10. However, as Fig. 10 shows, such behaviour disappears forω = 10, in which case α stays always positive and γ always negative. In fact, there is an asymptotic scaling α/u0 ∼ R −1/2 m as Rm → ∞, and γ approaches a constant finite value as Rm → ∞.
In a few cases ηt and δ have been determined in addition to α and γ. Results on the dependence of these quantities with ǫ = 1 andω = 0.7 on Rm are shown in Fig. 11. They have however been calculated with k = kH, not k → 0, and are therefore at most approximations of the mentioned quantities.
A correct interpretation of these results requires a look on the explanations of Sect. 4 on the non-local connection between E, B and J as defined by (46). In that sense the α, γ, ηt and δ in Fig. 11   may be understood as values of the functionsα(k),γ(k),ηt(k) and δ(k) at k = kH.
In the following, when writing α(k) or ηt(k), for example, we always meanα(k) orηt(k). In an earlier investigation with the Roberts flow under SOCA and with isotropic turbulence independent of SOCA (Brandenburg, Rädler & Schrinner 2008) it was found that α(k) and ηt(k) vary with k in a Lorentzian fashion like (1+k 2 /k 2 H ) −1 . However, for the GP flow Courvoisier (2008) found that α(k) at small k is extremely sensitive to the value of Rm. Fig. 12 shows that α(k) and γ(k) for Rm = 30,ω = 0.5, and ǫ = 1, approach the values given in Fig. 9 as k → 0. However, the magnitudes of ηt(k) and δ(k) become rather large as k → 0. It turns out that α is positive for k/kH > 0.4 and γ becomes smaller with increasing k. Remarkably, ηt(k) is negative for k/kH < 1, suggesting that magnetic field generation might be possible via a negative magnetic diffusion instability.
In order to check this possibility we have calculated the linear growth rates (50) Fig. 13 shows that λ± is almost entirely given by ±α(k)k. A neg-  ative diffusivity instability does not occur. It is important to realize that most of the small wavenumber modes, especially those with negative values of ηt, would never be realized. This is because in a system of given size, only the corresponding harmonics will have a chance to be excited, and of those only the ones with the largest growth rates will dominate. We should point out that the detailed variations of λ+ shown in Fig. 13 may not be accurate. In fact, this figure shows a maximum at k/kH ≈ 0.75, but direct simulations suggest that the fastest growth occurs for k/kH ≈ 0.5 with a grow rate of λ ≈ 0.23urmsk f . Nevertheless, this value is still compatible with Fig. 13.

Case (ii)
In case (ii) we have calculated α and γ under the assumptions on φx and φy introduced in Sect. 2.4. Fig. 14 shows results for Rm ranging from 0.1 to 100 and ǫ = π as functions ofω. In the limit of smallω the flow can be considered as stationary, that is, as a Roberts flow. Indeed in this limit the values of α agree well with those obtained for the Roberts flow (e.g., Rädler et al. (2002a), see also Appendix A). For large values ofω the values of α vanish for all Rm. For not too smallω and Rm there is no longer a noticeable variation of α with Rm, and α reaches a maximum atω ≈ 0.3. In the range 0.3 ≤ω ≤ 1 continuous flow renewal removes the tendency for α to diminish with growing Rm. Forω ≤ 0.2 the value of α remains strongly dependent on Rm and can still change sign. We have also calculated α with ǫ = 1 and Rm = 100 as a function ofω and found a qualitatively similar behavior as for ǫ = π. In this case it remains positive and is up to 50% smaller than for ǫ = π whenω < 1 and somewhat larger whenω > 1. In all cases we found, as expected, γ = 0 within error margins.

DISCUSSION
Our results for the flow of type (i) confirm the finding of CHT06 that both the α and γ coefficients depend sensitively on Rm and also on ǫ, and that even the signs of these coefficients may vary with these parameters. We have to add that α and γ depend also onω, or q =ω Rm, that is, on parameters connected with the frequency of the wobbling motion, which CHT06 fixed in a special way without commenting on it, and that they show similar variations with these parameters. We found however rather regular behaviors of α and γ for small and for large values of ǫ andω. It is sometimes considered as a rule that the sign of α is opposite to that the mean helicity of the fluid flow and its modulus is proportional to that of the mean helicity. There is however no general reason for that kind of relation between α and the kinetic helicity. We see only two limiting cases which allow simple statements on the sign of α.
Secondly it was found in SOCA under the same conditions, but in the limit q → ∞, that the sign of α for a flow with finite correlation time is opposite to that of ∞ 0 u(x, t) · (∇ × u(x, t − τ ) dτ ; see, e.g., again Krause & Rädler (1980). A relation of that kind between α and this integral can indeed be formally derived from the general relation (29) of  and applied to our specific situation. In case (i) the correlation time is however infinite and this integral does not converge. Although we know that u · (∇ × u) = −2u 2 0 kH we do not see how reliable conclusions could be drawn concerning the sign of α in the limit q → ∞. In case (ii) the integral is positive, and indeed only positive α have been observed.
Beyond the low and high conductivity limits, that is, for not too small or not too large values of q, even SOCA offers no simple general statements on the sign of α. In general α may take both positive and negative values.
For studying α and ηt with very simple flows it seems appropriate to consider flows of type (ii) rather than of type (i). In case (i) the results are influenced by the aforementioned circular motion of the flow pattern. As long as only α and ηt should be discussed there is hardly a reason to introduce such a motion. We see no natural interpretation of it and so no interpretation of the so caused γ and δ effects.
Recently Tilgner (2008) pointed out that a time-dependent flow of a conducting fluid can act as a dynamo even when steady flows which coincide with it at any particular time cannot. He demonstrated this with a Roberts flow modified by a drift of its pattern so that the velocity u satisfies relations like (2) and (3) with ϕx = −kHv d t and ϕy = 0, where v d is constant the drift velocity. Even if the intensity of the flow is too weak so that in the case v d = 0 no growing solutions of the induction equation with a given period in the z direction exist, such solutions may occur in an interval of some finite v d . Although this flow considered by Tilgner is in a sense simpler than the flows in our paper, it shows no longer the symmetries with respect to the z axis which we have utilized. As a consequence the relation between the mean electromotive force E and the mean magnetic field B is more complex. In particular (13) and (14) no longer apply. Nevertheless the question arises whether the effect of the time-dependence of flows observed by Tilgner occurs also in the examples investigated here. In case (i) the parameter ω could play the role of v d . The fact that the magnitude of α is larger for some finiteω than forω = 0, which can be seen in Figs 7 and 8, points in this direction.
One of the original motivations for looking at the GP flow was the fact that it is time-dependent and in that sense closer to turbulent flows than time-independent flows. However, as we have shown here, the dynamo properties of the GP flow cannot be compared in a meaningful way with analytic theories or with simulations that apply to isotropic turbulence. Nevertheless, as shown in this paper, an analytic theory for the α effect and other turbulent transport coefficients can be derived that matches numerical results in limiting cases.
Astrophysical flows can often neither be described by isotropic turbulence nor by wobbling two-dimensional flow patterns, but they are likely to contain aspects of both extremes. However, the present work highlights another aspect that may be of more general significance and concerns the turbulent transport properties in the presence of high-frequency time variability. This is not just a peripheral aspect of turbulence, but it is an additional property whose effects need to be understood more thoroughly. The situation is reminiscent of the modifications of mixing length theory in the presence of stellar pulsations (see, e.g., Gough 1977). In dynamo theory the issue of high-frequency time variability has only recently been addressed. One example concerns the nonlinear α effect where its time dependence has a striking effect on the behaviour of the mean field. In that example the temporal behaviour of the forcing function (delta-correlated or steady) determines the nonlinear asymptotic scaling behavior of the quenching function α(B) at low Rm. The early results of Moffatt (1972) and Rüdiger (1974) suggested a |α| ∼ |B| −3 behavior, but in more recent years Field et al. (1999) and Rogachevskii & Kleeorin (2000) found instead a |α| ∼ |B| −2 behavior, which seemed in conflict with the earlier results. However, the work of Sur et al. (2007) now shows that this is not just an artifact related to different approximations, for example, but it depends on whether or not the flow is timedependent. They found that the |α| ∼ |B| −3 behavior is reproduced if the flow is steady, while the |α| ∼ |B| −2 behavior is obtained in the time-dependent case using a forcing function that is δ-correlated in time. Again, it is not clear which types of flows are more astrophysically relevant, but it is now clear that the detailed time-dependence of the turbulent flows can affect its transport properties in rather unexpected ways.

APPENDIX A: ROBERTS FLOW
In the special case ǫ = 0 the Galloway-Proctor flow, defined by (2) and (3), turns into the Roberts flow. Our SOCA results for this special case agree with results for the Roberts flow reported in  and in Rädler et al. (2002a), referred to as BRS08 and R02a, respectively.
In BRS08 instead of our coordinate system (x, y, z) another one, say (x ′ , y ′ , z), is used, which is obtained by a 45 • rotation of our system about the z axis, that is, In the case ǫ = 0, to which we restrict ourselves here, (3) turns under this transformation into Together with (2) we find so, referring to the system (x ′ , y ′ , z), Comparing this first with (BRS08 25) and ignoring the opposite sign of uz we find The only consequence of inverting the sign of uz is a sign change of α. Taking then the SOCA results (BRS08 30) for α and ηt, with u BRS 0 in place of u0 and completed by (BRS08 29), considering (A4) and the remark on the sign of α we can easily reproduce our results (23).
Going beyond SOCA we note that according to (BRS08 25), or also according to (R02a 20), with a function φ satisfying φ(0) = 1 and vanishing like R −3/2 m with growing Rm. It has been calculated numerically and is plotted, e.g., in R02a.
The result given by (B8) and (B9) is valid for arbitrary k. This applies of course also if it is written in the alternative form with CC and CS. In that sense it is of some interest in view of the nonlocal connection between E and B studied in the paper by Brandenburg, Rädler & Schrinner (2008). In the main part of the present paper we consider however the limit k → 0 only. In this limit (B7) applies with k = 0. Then (B8) and (B9) agree just with (16) and (19).

APPENDIX C: HIGHER-ORDER CALCULATIONS
For the sake of simplicity we assume now, beyond SOCA, that B is a uniform field, that is, has no spatial derivatives. Then u × b is independent of space coordinates and (28) turns into We may apply some modification of the procedure used in Appendix B for solving the equation (B1) forb to the equations (C1) for b (2) and b (3) .