Small-Scale Kinematic Dynamo and Non-Dynamo in Inertial-Range Turbulence

We investigate the Lagrangian mechanism of the kinematic ``fluctuation'' magnetic dynamo in turbulent plasma flow at small magnetic Prandtl numbers. The combined effect of turbulent advection and plasma resistivity is to carry infinitely many field lines to each space point, with the resultant magnetic field at that point given by the average over all the individual line vectors. As a consequence of the roughness of the advecting velocity, this remains true even in the limit of zero resistivity. We show that the presence of dynamo effect requires sufficient angular correlation of the passive line-vectors that arrive simultaneously at the SAME space point. We demonstrate this in detail for the Kazantsev-Kraichnan model of kinematic dynamo with a Gaussian advecting velocity that is spatially rough and white-noise in time. In the regime where dynamo action fails, we also obtain the precise rate of decay of the magnetic energy. These exact results for the model are obtained by a generalization of the ``slow-mode expansion'' of Bernard, Gawedzki and Kupiainen to non-Hermitian evolution. Much of our analysis applies also to magnetohydrodynamic turbulence.


I. INTRODUCTION
Turbulent magnetic dynamo effect is of great importance in astrophysics and geophysics [1]. Many questions remain, however, about the basic mechanism of dynamo action, even for the kinematic stage when the seed magnetic field is weak and does not react back on the advecting velocity field. Stretching of field lines by a chaotic flow is, of course, the ultimate source of growth of magnetic field strength. Plasma resistivity η in turn acts to damp the magnetic field. However, the dynamo cannot be understood as a simple competition between growth from stretching and dissipation from resistivity. For example, resistivity plays also a positive role in dynamo effect through the reconnection of complex, small-scale field-line structure [2].
In addition, random advection may not lead to field growth in the limit of vanishing resistivity. Consider, for instance, the kinematic dynamo model of Kazantsev [3] and Kraichnan [4,5] with a Gaussian random velocity that is delta-correlated in time. In this model there is a dramatic dependence of dynamo effect on the spatial rugosity of the velocity, as measured by the scaling exponent 0 < ξ < 2 of the spatial 2-point velocity correlation [3]. There exists a certain critical value ξ * such that for ξ < ξ * , kinematic dynamo effect exists only above a threshold value P r c of the magnetic Prandtl number P r = ν/κ [6,7]. Here κ = ηc/4π is magnetic diffusivity * Electronic address: eyink@ams.jhu.edu † Electronic address: antfrannet@gmail.com while ν is an effective viscosity associated to a "dissipation length" ℓ ν of the velocity, above which scaling holds with exponent ξ and below which the synthetic field becomes perfectly smooth. In this regime of extreme roughness of the advecting velocity field, there is no kinematic dynamo even as κ, ν → 0, if P r < P r c .
On the contrary, in the Kazantsev-Kraichnan (KK) model for smoother velocity fields with ξ > ξ * there is a critical value Re m,c of the magnetic Reynolds number Re m = u rms L/κ, where u rms is the root-mean-square velocity and L is the integral length-scale of the fluctuating velocity. See [6,7], also [8]. In this smooth regime, small-scale kinematic dynamo leads to exponential growth of the rms magnetic field, even for P r → 0 as long as Re m > Re m,c . The most natural correspondence of the KK model to the kinematic dynamo problem in real fluid turbulence is for the value ξ = 4/3, which is greater than the critical value ξ * = 1 in three space dimensions. This correspondence would suggest that there is a critical magnetic Reynolds number for onset of kinematic dynamo in actual fluid turbulence, but no lower bound on the magnetic Prandtl number. On the other hand, numerical studies of Schekochihin et al. [9,10] suggested that hydrodynamic turbulence is instead like the rough regime of the KK model and that the critical Prandtl number P r c for small-scale dynamo action tends to a finite, positive value as Re m → ∞. Their latest investigations now support the opposite conclusion, that the critical magnetic Reynolds number Re m,c tends to a finite, positive value as P r → 0 [11,12]. Considerable debate still continues, however, about the precise nature and universality of the observed small-scale dynamo.
To resolve such subtle issues a better physical under-standing is required of the mechanism of the turbulent kinematic dynamo. In our opinion, important ideas have been contributed recently by Celani et. al. [13]. They pointed out that the existence of dynamo effect in the KK model for space dimension d = 3 should be closely related to the angular correlation properties of material line-vectors. They considered the covariance at time t of two infinitesimal line-vectors that are advected starting a distance r apart at time 0. Celani et al. argued that this correlation vanishes as r decreases through the inertial scaling range or as t → ∞, going to zero as a power (r/t 1/γ ) ζ . Here γ = 2 − ξ and ζ = ζ(ξ) is the scaling exponent of a "homogeneous zero-mode" for the linear operator M * 2 that evolves the pair correlations of line-elements forward in time. Celani et. al. [13] further claimed that the transition between dynamo regimes in the KK model for d = 3 corresponds exactly to the value ξ * = 1 where ζ(ξ * ) = 0 [13].
In this paper we shall further investigate these questions. In the first place, we shall show that the claims of Celani et. al. [13] are not quite correct. It will be shown here that the specific correlation function proposed by those authors does not discriminate between dynamo and non-dynamo regimes. The scaling law which they proposed is valid, but holds over the entire range 0 < ξ < 2 with a different zero-mode and different scaling exponent than they had claimed. We shall show that a quite different correlation function of material line-elements is necessary to serve as an "order parameter" for kinematic dynamo. The crucial difference is that the quantity introduced here measures the angular correlation of material line-vectors that are advected to the same space point at time t. But still, why should there be any connection of roughness exponent ξ with dynamo action? Individual field lines ought to be stretched and their field strengths increased for all values of ξ. We shall provide in this work a plausible physical explanation. Although individual lines may stretch due to chaotic advection, infinitely-many magnetic field lines will arrive at each point of the fluid due to diffusion by resistivity and the final magnetic field will be the average value that results from reconnection and "gluing" of field lines by resistivity. We shall show that too little angular correlation leads to large cancellations in this resistive averaging, with the net magnetic field suffering decay despite the growth of individual field lines.
We devote the remainder of our paper to a detailed study of the "failed dynamo regime" in the KK model for ξ < ξ * and P r < P r c . Part of our motivation is the speculation of [9,10] that hydrodynamic turbulence at high magnetic Reynolds number but low P r resembles this parameter range of the KK model. A better understanding of this regime may be useful to rule out its validity for hydrodynamic turbulence, based on astrophysical observations. We shall see, for example, that it implies a very rapid rate of decay of an initial seed magnetic field. Indeed, we show that in the KK model for rougher velocities the decay of the magnetic field is not resistively limited, with dissipation rate nonvanishing even in the zero-resistivity limit κ → 0, as long as P r < P r c . There is a strong analogy with the anomalous decay of a turbulence-advected passive scalar, for which scalar dissipation is non-vanishing even in the limit of zero scalar diffusivity [14,15]. The decay rate is instead determined by large-scale statistical conservation laws, associated to "slow modes" of the scalar evolution operator. We show here that the decay of the magnetic field in the rough regime of the KK model is determined in the limit κ → 0, P r < P r c by the "slow modes" of the linear evolution operator M * 2 for pairs of infinitesimal line-elements. We shall establish these results by a formal extension of the slow-mode expansion of Bernard et al. [16] to the case of non-Hermitian evolution operators, which is presented in the Appendix. We shall furthermore determine all self-similar decay solutions of the magnetic field in the non-dynamo regime of the KK model, following [14] for the passive scalar. Unlike the scalar case, however, determining the decay law of the magnetic energy requires an additional step of matching these self-similar solutions to explicit resistive-range solutions. We shall use these results to discuss the physical mechanism of kinematic dynamo, and, in particular, to relate our dynamo "order parameter" to the process of "induction" by a spatially uniform initial magnetic field. As we shall see, considerable insight can be obtained into the inner workings of the small-scale dynamo by considering also the situations where it fails.

A. The Kinematic Dynamo
The evolution of the passive magnetic field B(x, t) is governed by the induction equation where u ≡ u(x, t) is the advecting velocity field and κ is the magnetic diffusivity. The magnetic field is taken to be solenoidal, assuming there are no magnetic monopoles: Notice that this condition is preserved by the evolution equation (1) if it is imposed at the initial time t 0 = 0. We have also assumed above that the advecting fluid is incompressible so that For simplicity, we shall only consider this case hereafter.
For an incompressible fluid, one can represent the solution of the induction equation by a stochastic Lagrangian representation of the following form: See [13,17]. Hereã(x, t) are the "back-to-label maps" for stochastic forward flowsx(a, t) solving the SDE dx(a, t) = u(x(a, t), t)dt + √ 2κ dW(t).
E denotes average over the realizations of the Brownian motion W(t) in Eq. (5). For the ith component of the magnetic field we can write where we have defined These results may be used to represent the equal-time, 2-point correlation of the magnetic field, averaged over the stochastic velocity u and the random initial magnetic field B 0 : We have assumed that u and B 0 are statistically independent and we also have defined For statistically homogeneous velocity and initial condi- Here the prime ′ denotes a second Brownian motion W ′ (t) statistically independent of W(t). Equation (9) was introduced by Celani et al. [13] and heavily exploited in their analysis of the magnetic correlation. Another closely related propagator was introduced by [13] related to infinitesimal material line-elements, which evolve according to the Lagrangian equation of motion: Note that the positions of the line-elements are assumed to move stochastically according to (5), so these are not quite "material lines" in the traditional sense when κ > 0. The exact solution of the above equation for t > 0 is withx(a, t) solving (5). Taking initial line-elements δℓ i k (0) = δ i k , δℓ ′j ℓ (0) = δ j ℓ starting at positions a, a ′ displaced by r = a ′ − a, one may follow [13] to define for statistically homogeneous turbulence If we make the change of variables a → x in the argument of the delta function of eq.(10), then the jacobian of this transformation of variables is 1 due to incompressibility. Therefore, one finds by comparison with (11) that equating the two propagators under interchange of arguments. In our work below an important role will also be played by the covariant vector given by the gradient G = ∇θ of a passive scalar θ. The gradient satisfies the equation which is dual to the equation (1) for the contravariant vector B [18]. The above equation preserves the condition G = ∇θ if this is imposed at time t 0 = 0. A stochastic Lagrangian representation also exists for the solution of this equation. Solved forward in time with κ > 0 this representation involves the matrix ∇ xã (x, t). However, taking κ → −κ in (12) and solving backward from time t > 0 to time 0 yields the representation for the i component of the gradient field: For statistically homogeneous velocity and initial conditions we introduce the 2-point correlation of the gradient field, G ij (ρ, t) ≡ G i (a, 0)G j (a ′ , 0) with ρ = a − a ′ . By the same arguments as previously for positive times t > 0. We shall generally avoid using the geometric language of differential forms and Lie derivatives in this paper, but a few brief remarks may be useful. For those unfamiliar with this formalism, a good introductory reference is [19]. The magnetic field B discussed above is a 1-form, which is more properly represented by its Hodge dual B * , a (d − 1)-form. The equation (1) for κ = 0 is equivalent to ∂ t B * + LB * = 0, where L is the Lie derivative. The Lie derivative theorem thus implies that (1) for κ = 0 satisfies an analogue of the Alfvén theorem, with conserved flux of B through (d − 1)-dimensional material hypersurfaces. On the other hand, the field G is a proper 1-form and equation (12) for κ = 0 is equivalent to ∂ t G + LG = 0. The Lie derivative theorem thus implies that integrals of G along material lines are conserved for κ = 0. Either of these equations could be regarded as a valid generalization of the kinematic dynamo problem to general space dimension d. The non-gradient solutions G of (12) are generalizations of the 3-dimensional vector potential and the "magnetic flux" is represented by their line-integrals around closed material loops. For d = 3 the non-gradient solutions of (12) are in one-to-one correspondence (up to gauge transformations) with the solenoidal solutions of (1), by the familiar relation B = ∇×G.
We note finally that all of the results in this section hold for any random velocity field that is divergencefree and statistically homogeneous. Thus, they apply not only to the Kazantsev-Kraichnan model discussed in the following sections, but also to the kinematic dynamo problem for hydrodynamic turbulence governed by the incompressible Navier-Stokes equation.

B. White-Noise Velocity Ensemble
In the Kazantsev-Kraichnan (KK) model [3,4,5] the advecting velocity u(x, t) is taken to be a Gaussian random field with zero mean and second-order correlation delta function in time, given explicitly by Under the incompressibility constraint ∂ i S ij (r) = 0 and supposing S ij (r) scales as ∼ r ξ for ℓ ν ≪ r ≪ L, one deduces for that range that wherer i = r i /r. Define "viscosity" ν = D 1 ℓ ξ ν . Below we shall consider especially the limit ν, κ → 0 with ν < P r c κ. This is the non-dynamo regime in the limit of infinite kinetic and magnetic Reynolds numbers. One of our main objectives is to understand better the geometric and statistical properties of this regime which lead to the failure of small-scale dynamo action.
In addition to the properties of statistical homogeneity, stationarity and incompressibility, the white-noise velocity ensemble is time-reflection symmetric. This implies F ij kℓ (r, −t|ρ, 0) = F ij kℓ (r, t|ρ, 0) and the similar property for F ij kℓ (ρ, t|r, 0) . Combined with the other symmetries, this imples that The first line follows from incompressibility, the second line is due to time-translation invariance and the last equality follows from time-reflection symmetry. Thus the two propagators are adjoints in the KK model. Time-reflection symmetry has also an important implication for the evolution of the gradient field correlation. Note that time-translation invariance implies that equation (13) can be written for t > 0 as Then time-reflection symmetry implies further that for t > 0. Compare with equation (9) for the magnetic correlation. We see that the F -propagator in the KK model evolves forward in time the gradient correlation.
The most important property of the white-noise model is its Markovian character, which implies that timeevolution of correlations is governed by 2nd-order differential (diffusion) equations. E.g. the n-th order equal time correlation function C i1i2...in n ≡ n a=1 B ia (x a , t) satisfies an equation of the form ∂ t C n = M n C n . Expressions for the general n-body diffusion operators M n can be found in [20], which can be obtained using Itô formula as in Ref. [7] or, equivalently, by Gaussian integration by parts. In the limit ν, κ → 0 all of these operators for general n become degenerate (singular) and homogeneous of degree −γ with γ = 2−ξ. Below we shall mainly consider n = 2 and thus write simply M for M 2 . However, many of our considerations carry over to general n, as will be noted explicitly below. Following the notations of [13], we write for n = 2: with The notation M(r) indicates that ∂ i = ∂/∂r i . Note that equation (17) has an invariant subspace satisfying It follows from (17) and (9) that the F -propagator is the heat kernel of the adjoint operator satisfying The propagator F is thus the heat kernel of M: Because of the homogeneity of the operators M and M * in the ν, κ → limit, F satisfies the scaling relation with an identical relation for the F -propagator. Finally, it follows from (16) that the gradient correlation satisfies This equation has an invariant subspace of solutions of the form G ij = −∂ i ∂ j Θ for a scalar correlation function Θ(r, t). Celani et al. [13] have also introduced the quantity where on the righthand side the line-elements are initially unit vectors δℓ k (0) =ê k , δℓ ′ ℓ (0) =ê ℓ starting at positions displaced by r. This quantity measures the angular correlation of the material line-elements at times t > 0, as well as their growth in length. It follows from (21) that this quantity in the KK model satisfies with initial condition Q kℓ (r, 0) = δ kℓ , as already noted in [13]. This equation is identical to (23) for the gradient correlation and, furthermore, Q kℓ (r, 0) = −∂ k ∂ ℓ Θ(r, 0) with Θ(r, 0) = −(1/2)r 2 . Thus, Q is of gradient type. In this work we restrict ourselves to conditions of statistical homogeneity, isotropy and parity invariance for all stochastic quantities. Thus, we can write the 2-point correlation function of the magnetic field as wherer i = r i /r. C L and C N are the longitudinal and transverse correlations, respectively. With the form of the velocity correlation in (15), the evolution equation (17) reduces to two coupled equations for C L and C N . A lengthy but straightforward calculation gives and respectively, when ν, κ → 0. For solenoidal solutions, such as for the magnetic field, it is easy to show that the two correlations are related by For example, see [18,21]. The solutions satisfying this relation form an invariant subspace, with the evolution reducing to a single equation for C L : In the same manner, the general solution of (23) or (25) may be decomposed as under assumptions of homogeneity, isotropy and reflection-symmetry. Then G N and G L satisfy the following coupled equations for ν, κ → 0 and respectively. Gradient solutions satisfy the constraint where G N = − 1 r ∂ r Θ in terms of the scalar correlation function Θ. In this invariant subspace of solutions the dynamics reduces to a single equation for G N :

C. Line-Vector Correlations
Kinematic dynamo effect is due ultimately to the stretching of magnetic field lines as they are passively advected by a chaotic velocity field. However, the properties of infinitesimal material line-elements in the KK model are, at first sight, counterintuitive in this respect. In order to discuss stretching of individual lines, it must be understood that the velocity field is smoothed at very small scales ℓ ν . The inertial-range velocity structure function in (15) crosses over to a viscous-range form for r ≪ ℓ ν . The growth of line-elements in such a smooth velocity field, white-noise in time was derived by Kraichnan [22] to be exponential with the Lyapunov exponent See also [23,24,25]. The "material" line-elements of relevance to the kinematic dynamo are subject to an additional Brownian motion proportional to √ 2κ in (5). However, this effect of molecular diffusivity κ corresponds just to changing the constant D 0 in the Kraichnan-model velocity covariance (14) to D 0 + 2κ. Since only velocitygradients enter (38), this result for the Lyapunov exponent still holds in the presence of molecular diffusivity.
It follows from (38) that line-stretching is greater for smaller ξ and smaller ν. This may seem a bit perplexing, because the dynamo fails for ξ too small, in the range 0 < ξ < ξ * . In that regime, there is no dynamo action for ν < P r c κ, despite the fact that the stretching rate becomes larger as ν decreases. The turbulent kinematic dynamo cannot be understood as a simple "competition" between stretching and diffusion. What, then, can account for the presence of the dynamo in the range ξ * < ξ < 2 of smoother velocities in the KK model and failure of the dynamo in the range 0 < ξ < ξ * of very rough velocities? An intriguing suggestion has been made by Celani et al. [13] that the existence of dynamo effect can be characterized by the angular correlations of material line-elements. They proposed the function Q defined in (24) as an "order parameter" for the dynamo transition. As we shall demonstrate below, the principal conclusions of [13] about Q are erroneous and this quantity does not discriminate between dynamo and non-dynamo regimes in the KK model. However, our discussion will lead us to identify a different correlation property of infinitesimal material line-vectors, which can indeed serve as an "order parameter" for the dynamo.
The principal claims of [13] were as follows. First, in the non-dynamo regime for 0 < ξ < ξ * with rough velocity, Q exhibits a scaling law of correlations: for r ≪ min{(D 1 t) 1/γ , L}. For finite ν, κ, this relation holds in the inertial-convective range of scales max{ℓ κ , ℓ ν } ≪ r ≪ L, with ℓ κ = (κ/D 1 ) 1/ξ (assuming that ν < P r c κ). This is an example of "zero-mode dominance" [16]. Thus, the quantity Z kℓ (r) is a homogeneous zero mode of the operator M * , satisfying [M * ] pq kℓ Z pq = 0, with exponent ζ(ξ) > 0 for 0 < ξ < ξ * . Intriguingly, it was found that ζ = ζ + 2, where ζ is the scaling exponent of the zero mode of M which was shown in [26] to dominate in the magnetic correlation of the KK model for the same parameter range. It was furthermore claimed in [13] that ζ(ξ * ) = 0. For 2 > ξ > ξ * , on the contrary, it was argued that M * develops point spectrum and that where E 0 is the largest positive eigenvalue of M * and E kℓ (r) is the corresponding eigenfunction. E 0 is numerically equal to the dynamo growth rate. Notice in the limit P r ≪ 1 that E 0 ∝ 1/t κ = (D 2 1 /κ γ ) 1/ξ [6], so that κ must be kept nonzero (but with Re m > Re m,c ). Thus, the "material lines-vectors" in Q are advected by velocity u subject to Brownian noise proportional to √ κ. The appropriate terms proportional to κ must then be included in the diffusion operators M and M * [13]. We shall show that the zero-mode dominance relation (39) does hold for Q but with a different zeromode and different scaling exponent ζ than that claimed by [13]. Furthermore, the scaling relation (39) holds over the whole range 0 < ξ < 2, assuming only that max{ℓ κ , ℓ ν } ≪ r ≪ L, with an exponent ζ(ξ) = −ξ < 0 which does not exhibit any qualitative change at the dynamo transition ξ = ξ * . The exponential growth relation (40) does not hold for the quantity Q anywhere over the range 0 < ξ < 2, even if P r > P r c and Re m > Re m,c .

D. Zero-Mode Analysis
The basic tool of our investigation is a generalization of the slow-mode expansion of Bernard et al. [16]. Those authors derived such an expansion for the propagator or heat kernel P (r, t|r 0 , t 0 ) that describes the evolution of a passive scalar in the Kraichnan white-noise velocity ensemble with covariance (14). However, the derivation of [16] was, in fact, axiomatic and applicable to the propagator for any non-positive, self-adjoint operator, with absolutely continuous spectrum and homogeneous of degree −γ. This derivation showed that the slow mode expansion follows from assumed meromorphic properties of the Mellin transform of the propagator and Green's function of the operator. In the Appendix of this paper, we generalize this axiomatic derivation to the case of the non-Hermitian operators M and M * , in the non-dynamo regime where both have absolutely continuous spectrum. We refer the reader to the appendix for details and here just state the essential results.
The operators M and M * have two types of homogeneous zero-modes, regular and singular. The regular zero modes are denoted Z (a) and Z (a) for a = 1, 2, 3, ..., respectively, with exponents ζ a and ζ a whose real parts increase with a. These are ordinary functions which satisfy the conditions MZ (a) = 0 and M * Z (a) = 0 globally.
The singular zero modes W (a) and W (a) for a = 1, 2, 3, ..., instead have exponents ω a and ω a whose real parts decrease with a for a = 1, 2, 3, ..., respectively. These are distributions which satisfy the conditions MW (a) = 0 and M * W (a) = 0 only up to contact terms. The scaling exponents of the two sets of zero modes are related by Above each regular zero mode lies an ascending tower of slow modes Z (a,p) and Z (a,p) homogeneous of degree ζ a,p = ζ a + γp and ζ a,p = ζ a + γp, respectively, satisfying MZ (a,p) = Z (a,p−1) and M * Z (a,p) = Z (a,p−1) for p = 1, 2, 3, ... with Z (a,0) = Z (a) and Z (a,0) = Z (a) . In addition, below each singular zero mode is a descending tunnel of self-similar decay solutions W (a,p) (r, t) and W (a,p) (r, t) These are related to the singular zero modes by MW (a,p−1) = In terms of these quantities there are short-distance expansions, for λ ≪ 1, both for the F -propagator and for the F -propagator See the Appendix for the details of the derivation. Note that these asymptotic series are generally dominated by their leading terms for a = 1 and p = 0, corresponding to the regular zero mode with scaling exponent of smallest real part. Of course, the leading term may give a zero contribution for various reasons and then subleading terms will dominate instead. In order to make use of this expansion we must calculate explicitly the homogeneous zero modes of M and M * .
To find the isotropic and scale-invariant zero-modes of M we substitute into (27) and (28) the forms giving the matrix equation Calculating the determinant one finds that the scaling exponents σ are Note that the set ζ 1 , ω 2 correspond to the invariant subspace of solenoidal solutions, as may be verified by substituting the scaling ansatz for C L into (30). The exponent ζ 1 coincides with that found by Vergassola [26] to dominate in the magnetic 2-point correlation for a forced steady-state at high magnetic Reynolds number and zero Prandtl number.
To find the isotropic and scale-invariant zero-modes of M * we likewise substitute into (32) and (33) the forms giving the matrix equation Calculating the determinant one finds that the scaling exponents σ are Note that the set ζ 1 , ω 2 corresponds to the invariant subspace of gradient solutions, as may be verified by substituting the scaling ansatz for G N into (35). These exponents are directly related to those for the passive scalar discussed in [16]. In the isotropic sector of the scalar, the regular zero mode has exponent 0 (constants) and the singular zero mode has exponent γ − d. The first vanishes after taking gradients and is replaced by its lowest-lying slow mode, with exponent γ. Taking two space derivatives reduces these exponents by 2, The exponents ζ 2 , ω 1 coincide (for d = 3) with the set which were claimed in [13] to determine the scaling of Q.
It is easy to check the relations for a = 1, 2, consistent with the general result (41). We see that, at least in the isotropic sector, the solenoidal scaling solution W (2) (r, t) is associated in the slow-mode expansion to the non-gradient zero-mode Z (2) (r) and the gradient scaling solution W (2) (r, t) is associated to the non-solenoidal zero-mode Z (2) (r). In fact, this is true in general, as we now show. Take any solenoidal scaling solution W (a,p) (ρ, t). Then it follows from the propagator relation (9) and the scaling property of the singular slowmode W (a,p) (r, 0) that This can only be consistent with the slow-mode expan- Since W (a,p) is solenoidal, then Z (a,p) must be nongradient. Otherwise the integral will vanish, because the solenoidal and gradient subspaces are orthogonal. An identical argument shows likewise that any gradient scaling solution W (a,p) is associated in the slow-mode expansion of F to a non-solenoidal zero-mode Z (a,p) .
As should now be clear, however, the relation (39) cannot hold with ζ = ζ 2 and Z = Z (2) . Since Q is of gradient type, its evolution is described by (35) which has Z (1) as its only regular zero mode with scaling exponent ζ 1 . We shall now verify this directly from the definition (24) of Q, by means of the slow-mode expansion. We use first the adjoint relation F ij kℓ (ρ, t|r, 0) = F ij kℓ (r, t|ρ, 0) and the homogeneity relation (22) for F to write Notice, however, that the space integral vanishes for all W (a,p) in the solenoidal sector (e.g. see [6]). This follows for any solenoidal correlation C, from where A kℓ is the correlation of the vector potential A. Note that, in general dimension d, B is a 1-form and A is a 2-form, related by the codifferential B = δA [40].
Since the solenoidal solutions W (a,p) are associated in the expansion to the slow modes Z (a,p) outside the gradient sector, all of these terms drop out in Q. The result is the same as the slow-mode expansion carried out entirely in the gradient sector, with the leading term . This is the correct relation replacing relation (39) claimed in [13].
This same relation may be verified by appealing to the results of Eyink and Xin [14] on the self-similar decay of the passive scalar. Those authors found that there is a universal form of the self-similar decay solutions for the scalar correlation function at short distances: [14], equation (3.21). Here For general initial data with power-law decay of correlations in space, Θ(r, 0) ∼ r −α , the decay rate is given by ϑ 2 (t) ∼ t −α/γ at long times [14]. Since Q kℓ (r, 0) = δ kℓ corresponds to Θ(r, 0 The latter result may be verified from equation (34) by substituting G N = A N r −ξ . This alternative derivation of (45) makes clear its validity over the whole range 0 < ξ < 2 and not just 0 < ξ < ξ * . There can be no exponential growth relation for Q, such as relation (40) proposed in [13]. It is true that the operator M * must have a positive eigenvalue whenever M does so. However, the corresponding eigenfunctions must lie in the non-gradient sector. An exponential growth for Q kℓ (r, t) = −∂ k ∂ ℓ Θ(r, t) would require an exponential growth for the scalar correlation function Θ(r, t), which does not occur.

E. A Dynamo Order-Parameter
Based on the previous discussion, we now will propose an alternative definition of a line-correlation which can serve as an "order parameter" for the dynamo transition. Clearly, one should not integrate F ij kℓ (ρ, t|r, 0) over ρ, as this eliminates the solenoidal sector. We propose instead to set ρ = 0, defining: where, as before, the two line elements are started with Because of the delta-function, R kℓ (r, t) measures the growth in magnitude and the angular correlation between those material line-vectors which arrive, stretched and rotated, at the same point at time t. Just like the quantity Q defined in [13], R satisfies also the equation However, it has the initial value which is non-gradient, unlike for Q. Thus, R should experience exponential growth like (40) in the dynamo regime for 2 > ξ > ξ * .
The time-dependence in the non-dynamo regime for 0 < ξ < ξ * can be obtained from the slow-mode expansion of F ii kℓ (λr, 1|0, 0) with λ = r/(D 1 t) 1/γ . One obtains for λ ≪ 1 that Thus, R exhibits a power-law decay in time, with the dominant terms given by the two isotropic zero modes In all dimensions d the exponent ζ 2 > 0 for 0 < ξ < 1, becoming negative for ξ > 1. Thus, the first term with ζ 1 = −ξ dominates for lower ξ values. There is a critical dimension d c . = 4.659, given by the positive real root of the cubic polynomial d 3 − 8d 2 + 19d − 16, above which it instead true that ζ 2 < ζ 1 when ξ > ξ c with Note that ξ c < ξ * where the dynamo transition occurs. The latter value [7] is the point at which ζ 2 develops an imaginary part and the slow-mode expansion above breaks down. For exponents ξ * < ξ < 2 in all the integer dimensions 2 < d < 9, the power-law decay is replaced by exponential growth proportional to the eigenfunction E kℓ (r) of M * with largest eigenvalue E 0 . To demonstrate this, it is enough to show that the initial condition R kℓ (r, 0) gets a nonzero contribution from the eigenfunction E kℓ (r). We may represent this initial state by an expansion where E α kℓ (r) is the eigenfunction of M * with eigenvalue E α . Note that for the continuous spectrum, this is a generalized eigenfunction expansion where the sum over α is a continuous integral and E α kℓ are distributions, not square-integrable functions. The expansion coefficients are given by where E kℓ α are the eigenfunctions of M with the same eigenvalue E α . These form a bi-orthogonal set with the eigenfunctions E α kℓ of M * [41]. The coefficient C 0 corresponding to the eigenfunction E 0 kℓ = E kℓ is non-zero because where E kk (0) = 0 is (twice) the energy in the normalized dynamo state.
Thus, unlike the quantity Q proposed in [13], the linecorrelation R defined in (47) satisfies the exponential growth relation (49) in the dynamo regime and powerlaw scaling (48) in the non-dynamo regime. It would be of great interest to determine the spatial structure of the eigenfunction E kℓ (r). Of course, this function must be of non-gradient type. It is known [6,8] that the trace of the dual eigenfunction E(r) = E ii (r) exhibits stretched-exponential decay of the form E(r) ∝ − exp −β(r/ℓ κ ) γ/2 for r ≫ ℓ κ . A similar behaviour for E(r) = E kk (r) can be checked to be consistent with the dynamical equations, but a more careful investigation is required. This will be pursued elsewhere.
A quantity with even simpler geometric significance which might also serve as an "order parameter" is This is the covariance of two material line-elements which started at any relative positions as identical unit vectors at time 0 and which ended at the same point at time t. The notation · 0 denotes the conditional expectation over material lines which end at zero separation. Clearly R(0) = 1. However, its time-dependence is undetermined by our present considerations both in the dynamo and in the non-dynamo regimes. We cannot argue that R(t) decays as a power in the non-dynamo regime, because the slow-mode expansion applies only for r ≪ (D 1 t) 1/γ whereas the definition of R(t) involves an integral over all r. We also cannot conclude that R(t) grows exponentially in the dynamo regime, because this requires the condition d d r E kℓ (r) = 0, which needs to be shown. However, we shall see in the next section that R(t) has a direct interpretation in terms of the turbulent decay of an initially uniform magnetic field and we shall determine its time-dependence in the non-dynamo regime.

III. DECAY OF THE MAGNETIC FIELD
We now consider in detail the problem of the turbulent decay of the magnetic energy B 2 (t) in the non-dynamo regime of the KK model, for 0 < ξ < ξ * and P r < P r c .

A. Discussion of the Convective-Range Decay Law
We begin by giving a simple, heuristic explanation of the decay law of the magnetic field, for generic initial data of the magnetic field with rapid decay of correlations in space. The fundamental observation is that the zeromodes Z (a) , a = 1, 2, 3, ... of the adjoint operator M * give rise to statistical conservation laws in the evolution of the 2-point correlation, which satisfy Note, however, that only the non-gradient zero modes lead to non-trivial conservation laws, because of the orthogonality of solenoidal and gradient correlations. The leading-order zero-mode is thus Z (2) found in the previous section, namely, This zero-mode coincides with that found by Celani et al. [13] for d = 3. The corresponding conserved quantity J 2 (t) plays the same role in the turbulent decay of the magnetic field as played by the "Corrsin invariant" in the decay of the passive scalar [14,15]. Assume, in fact, a self-similar decay law for the magnetic correlation C ij (r, t) = h 2 (t)Γ ij (r/L(t)).
The length L(t) is a large-distance correlation length or "integral length" of the magnetic field. The quantity h (t) is a measure of the magnitude of the magnetic fluctuations at scale L(t), which we term the magnetic amplitude. Just as for the scalar, the growth of the magnetic length-scale L(t) can be obtained dimensionally from yielding To determine the decay rate requires a relation between h (t) and L(t) which is provided by invariance of J 2 : with C = d d ρ Z (2) ij (ρ)Γ ij (ρ). Thus, finally, The generic decay of the magnetic amplitude is predicted by this argument to be determined by the scaling exponent ζ 2 , which decreases with increasing ξ over the range 0 < ξ < ξ * . Thus, the decay rate is faster for rougher velocities and slower for smoother velocities. It is noteworthy that the decay law (54) is completely independent of the resistivity. The above argument does not apply if J 2 = 0. In that case, one can expect that invariants from higher-order zero modes and slow modes of M * (again in the non-gradient sector) will determine the decay rate . Note, for example, that (d/dt)J 2,1 (t) = J 2 (t), so that J 2,1 becomes invariant if J 2 = 0. In ref. [14] it was shown that there are two universality classes in the turbulent decay of the passive scalar for generic initial data with rapidly decaying correlations in space, depending upon whether the "Corrsin invariant" J 0 from the constant zero mode is vanishing or nonvanishing. If J 0 = 0, then there exists a higher-order invariant J 1 = 0, associated to the first slow mode r γ in the tower above the constant zero mode, which determines the decay. Chaves et al. [15] showed how this picture emerges from the slowmode expansion of Bernard et al. [16] and extends to the higher-order correlations of the scalar. In the following section we shall present a similar treatment of the turbulent decay of the magnetic field, based on our generalized slow-mode expansion in the Appendix.
There are essential differences, however, between the turbulent decay of a passive scalar and of a passive magnetic field. Whereas the scalar field has a finite limit as diffusivity κ → 0, this is not true for the magnetic field which, even in the non-dynamo regime, tends to accumulate at the resistive scale [26]. As we shall see below, the scaling function Γ ij (ρ) grows with ρ decreasing through the convective range. Thus, one cannot set ρ = 0 to interpret h 2 (t) as the magnetic energy. A more correct interpretation of the magnetic amplitude is that h 2 (t)/L 2 (t) ≃ |A(t)| 2 , where A is a vector potential (2-form) such that B = δA. The decay rate of the magnetic energy |B(t)| 2 cannot be obtained from purely ideal considerations, but requires an explicit matching of convective-range solutions with resistive-scale solutions. In the following two sections we treat first the ideal, convective range problem with κ → 0.

B. Self-Similar Decay for Initial Data with Short-Range Correlations
Consider any initial 2-point correlation function C ij (r, 0) of the magnetic field which decreases rapidly for large r. We shall demonstrate that the correlation C ij (r, t) at much later times exhibits self-similar decay and determine the decay law. We use the propagator relation (9) and the symmetry properties of F to write: with λ = 1/(D 1 t) 1/γ and r = r/(D 1 t) 1/γ . In the last line we used the scaling property (22) for F . Since C kℓ (ρ, 0) decays rapidly for ρ ≫ L(0), we may employ the slowmode expansion (43) for (D 1 t) 1/γ ≫ L(0). Because C kℓ (ρ, 0) is solenoidal, only the non-gradient zero-modes of M * give a non-vanishing contribution. The leading-order term, in general, is In the last line we have used the self-similarity property We have also used the fact that W (1) is a real-valued function. This will be demonstrated in the following section, where we shall derive the explicit functional form of all the self-similar decay solutions. We conclude that, as long as J 2 (0) = 0, then the generic magnetic correlation C ij (r, t) with short-range initial data is proportional at long times to the self-similar decay solution W ij (1) (r, t). It is important to demonstrate that the above scenario is statistically realizable [14]. We shall construct now a positive-definite covariance function for which J 2 (0) = 0. This will also demonstrate the positive-definiteness of the scaling solution W (1) , since the dynamics is realizabilitypreserving and the above argument shows that lim λ→∞ λ d+ζ 2 C(λr, λ γ t) = J 2 (0) · W (1) (r, t).
As a simple example we take, with N = (σ/ √ 2π) d , A bit of calculation shows for this example that where S d−1 = 2π d/2 /Γ d 2 is the hypersurface area of the unit sphere in d-dimensions and A (2) L , A (2) N are the coefficients given in the previous section. At generic values of d and ξ, J 2 (0) = 0. It is noteworthy that J 2 (0) = 0 in the example above precisely at the point of degeneracy of zero-modes where ζ 1 = ζ 2 . As discussed in section II C, this occurs for d > d c . = 4.659 at the single value ξ = ξ c < ξ * . In fact, J 2 (0) = 0 at this point for all initial data, because there is then a single zero-mode of gradient-type satisfying A (2) L = (ζ 2 + 1)A (2) N ; see (34). Of course, whenever J 2 (0) = 0 then higher-order terms in the slow-mode expansion become dominant and a different self-similar solution W (a,p) (r, t) becomes the longtime attractor. We shall defer to future work the study of this non-generic situation.
In the remainder of this section we shall make some important comments about the generic case J 2 (0) = 0. Our first observation is about the property of "quasiequilibrium". It was shown in [14,15] that the shortdistance scaling of the scalar structure function in the decay of the passive scalar is identical to the scaling of the scalar structure function in a forced steady-state. This is the property of turbulence decay traditionally termed "quasi-equilibrium." We show here a similar property for the turbulent decay of magnetic field, using the slowmode expansion, as in [15] for the scalar. We use the propagator F, its scaling property (22), and the change of variables ρ = λρ with λ = 1/( We now employ the slow-mode expansion (42) of F for r ≪ (D 1 t) 1/γ to conclude that The scaling exponent ζ 1 and zero-mode Z (1) are the same as found in [26] to determine the short-distance scaling of the magnetic correlation function in the forced steadystate, which is just the "quasi-equilibrium" property [42]. Since ζ 1 < 0 for all 0 < ξ < ξ * , we see that C ij (r, t) increases without bound as r decreases, in agreement with our earlier physical discussion. We shall confirm this result by an independent argument in the next section. A second observation is that the above discussionthe demonstration of self-similar decay and quasiequilibrium-carry over directly to the general n-point correlation function of the magnetic field. Note that where r = (x 1 , x 2 , . . . x n ) and F n is the n-body propagator. All the symmetries used in the previous argument hold for general n, e.g., time-reversal and F n (ρ, t r, 0) = F n (r, 0 ρ, t). Note that due to space homogeneity only the separation of the variables matter (the absolute position of each particle is irrelevant) and we can work in the (n − 1)d-dimensional separation-of-variables sector. F n then has the scaling property F n (λρ, 0 λr, λ γ t) = λ −d(n−1) F n (ρ, 0|r, t) .
Finally, slow-mode expansions like (42) and (43) are valid for F n and F n for all integers n. See [16] and the Appendix for details. The whole analysis thus goes through as for n = 2 above and as in [15] for the scalar case.

C. General Self-Similar Decay
To complement the previous discussion employing the slow-mode expansion we shall here determine all possible self-similar decay solutions for the magnetic correlation function C, following the analysis in [14] for the passive scalar. It is convenient to employ the longitudinal correlation C L which satisfies the equation (30). We introduce the self-similar ansatz Substituting the ansatz (58) into equation (30) for C L we arrive at, with ρ = r/L, This implies that with constants α and β. We have freedom in choosing the value of β to fix the length-scale; here we adopt β = 1.
Making the substitution x = −ρ γ /γ(d − 1) yields An equation of this form can be solved by the Frobenius method (e.g. see [27], Sec.4.2). According to the general theory, there are two independent solutions of the form Γ(x) = x b Φ(x), where b is a root of the indicial equation If the two roots are distinct and do not differ by an integer, then the two functions Φ are both analytic, given by convergent power series. Otherwise, only one solution must be analytic and the second may be an analytic function plus C ln x times the first. In our case, it is easy to check that the roots of the indicial equation are just given by b = ζ 1 /γ, ω 2 /γ in terms of the scaling exponents of the zero-modes of M. If we substitute Γ = x ζ1/γ Φ into the equation for Γ, we obtain the Kummer equation [28] xΦ Both independent solutions can be obtained from this equation. The first is the Kummer function Φ(a, c; x), an entire function given by the power series with (a) n = a(a + 1) . . . (a + n − 1). The other is the Kummer function of the second kind, Ψ(a, c; x), which is defined by a suitable linear combination of Φ(a, c; x) and x 1−c Φ(a − c + 1, 2 − c; x). See [28], 6.5.6. It is not hard to check that this second term corresponds to the root b = ω 2 /γ of the indicial equation. However, we can argue as in [26] that matching the solutions in the convective range with those in dissipation range permits only the regular zero mode as an admissible physical solution. Thus, we obtain Γ(x) = x ζ1/γ Φ(a, c; x). This result can be simplified somewhat by appealing to the relation which follows by combining ζ 1 +ω 2 = −d−2ξ/(d−1) and ω 2 +ζ 2 = −d+γ. Note that the above relation generalizes the result ζ 2 = ζ 1 + 2 for d = 3 found in [13]. With this relation we obtain c = (ζ 1 + ζ 2 + d)/γ, so that All the self-similar solutions of Eq.(30) are given by the ansatz (58) with a scaling function of the form in (66) above and with L(t) and h (t) given by Eqs. (52) and (62), respectively. Since Φ(0) = 1 is finite, all of these self-similar solutions satisfy the condition of "quasiequilibrium," showing the same scaling r ζ1 for r ≪ L(t) as found in [26] for the forced steady-state. There are two distinct types of self-similar decay solutions corresponding to different choices of the free parameter α. When α = ζ 2 + d + pγ, for p = 0, 1, 2, . . ., then a = c + p with p = 0, 1, 2, . . .. In this case where L c−1 p (x) is the generalized Laguerre polynomial of degree p. (See [28], Ch.6). This series of solutions has stretched-exponential decay in space. If, for example, we take α = ζ 2 + d corresponding to p = 0, then we get The corresponding self-similar decay solution satisfies C(λr, λ γ t) = λ −(d+ζ 2 ) C(r, t). The α = ζ 2 +d solution thus coincides with the self-similar solution W (1) (r, t) which was shown in the previous section to describe the longtime decay of generic initial data with short-range correlations. More generally, the solutions with α = ζ 2 +d+γp coincide with the self-similar solutions W (1,p) (r, t) for p = 0, 1, 2, . . . which appear in the slow-mode expansion (43) of the adjoint propagator F . For any other choice of α = ζ 2 + d + γp with p = 0, 1, 2, . . . one obtains instead a class of self-similar decay solutions with power-law decay of correlations at large distances. This follows from the asymptotic relation Φ(a, c; −x) ∼ Γ(c) Γ(c−a) x −a for Re x → +∞, if a = c + p, p = 0, 1, 2, . . .( [28], 6.13.1). Using the above relation together with (66), (58), and (62) gives for any self-similar solution with α = ζ 2 + d+ γp, p = 0, 1, 2, . . . , where A is a time-independent constant. This result is usually called the "permanence of the large-scale eddies" in the turbulence literature. Note that for initial data with such power-law decay of correlations, the relation between h (t) and L(t) that determines the decay rate is obtained from this permanence, as h 2 (t) ≃ A[L(t)] −α , in agreement with (62). See [14] for more discussion.

D. Decay Law of the Magnetic Energy
We are now ready to discuss the decay law for the magnetic energy: Under the assumption of isotropic statistics made here, E(t) = (d/2)C L (0, t). Clearly, in order to evaluate this expression at r = 0, we must consider the matching of our convective range solution to the resistive scales. We may do this heuristically, as follows. We assume that, to leading order, We then estimate the correlation function on the right by matching with the convective-range expression for r ≪ L(t) and some positive constant C 0 . This yields Although the energy magnitude increases as resistivity is lowered, the decay rate is independent of resistivity. Since ζ 1 < 0, we see that the decay of magnetic energy E(t) ∝ (t − t 0 ) −(α+ζ1)/γ is always slower than the decay of magnetic amplitude h 2 (t) ∝ (t − t 0 ) −α/γ . For example, for the generic case with α = ζ 2 + d we obtain that The above argument is basically correct, but not fully rigorous. It seems worthwhile to give a more systematic derivation, since the time-dependence of magnetic energy is crucial to the issue of whether dynamo action is present or not. We employ a standard method of matched asymptotic expansions (see, for instance, Chap. V of Ref. [29]). The equation obeyed by C L for κ > 0 is See [7]. Substituting the self-similar ansatz (58) we get (68) with ρ = r/L(t) and ǫ ≡ ℓ κ /L(t). In the outer range where ρ = O(1), the dominant balance in the equation (68) is between the first term from the time-derivative, which acts like a forcing, and the second term from the turbulent advection. The third term may be neglected for small ǫ , yielding the leading-order equation for the outer solution. This is the same equation which was examined in the preceding section III C, where all self-similar solutions were found. Thus the outer solutions Γ out (ρ) are given by the formula (66) for any choice of α and multiplied by an arbitrary constant C out . These solutions must now be matched to an appropriate inner solution in the resistive range.
We introduce the inner variable σ = r/ℓ κ ≡ ρ/ǫ in Eq. (68) to obtain (69) The dominant balance in (69) is between the second term from the turbulent advection and the third term from the molecular resistivity. To leading order we can disregard the first term proportional to ǫ γ to get Making the change of variables y = −(d − 1)σ ξ reduces this to a hypergeometric equation ( [28], Ch.II): Up to an overall multiplicative constant, there is a unique solution F (a * , b * ; c * ; y) of the above equation which is analytic in the region arg(1 − y) < π of the complex yplane. This hypergeometric function is given for |y| < 1 by the absolutely convergent power series, if c * = 0, −1, −2, . . .. The other independent solution, y 1−c * F (a * + 1 − c * , b * + 1 − c * ; 2 − c * ; y) [[28], 2.9(17)], is singular at y = 0 and must be discarded. Because of the symmetry F (a * , b * ; c * ; y) = F (b * , a * ; c * ; y) we have freedom in choosing a * and b * . Combining the equations in (72) yields a quadratic equation for a * and an identical equation for b * . It is easy to check that the roots are just −ζ 1 /ξ and −ω 2 /ξ, where ζ 1 , ω 2 are the scaling exponents found in section II C. We choose a * = −ζ 1 /ξ and b * = −ω 2 /ξ. Thus, we obtain for the inner solution, with an arbitrary constant C in . This solution gives the complete description in the resistive range, e.g. implying a magnetic energy spectrum E(k) ∝ k −(1+ξ) for ℓ κ k ≫ 1.

E. Magnetic Induction and Dynamo Order Parameter
The above arguments are reminiscent of our discussion in subsection II C, where we emphasized the importance of considering the correlations between line-vectors advected to the same point, in order to distinguish between dynamo and non-dynamo regimes. In fact, the two subjects are intimately related. As we now show, the "order parameter" R(t) that we considered in (50) can be interpreted as the energy of a certain self-similar decay solution C (0) corresponding to an initial condition which is a random, statistically isotropic but spatially uniform magnetic field. Such a random magnetic field has a covariance of the form for a positive real number A. A constant correlation such as above would be invariant for an advected scalar, but it is not for a magnetic field. There is a well-known physical phenomenon of "shredding" [1] or "induction" [12] of a constant magnetic field due to the stretching term (B·∇)u in the evolution equation. Thus, an initially constant magnetic field will develop very fine-scale structure by turbulent induction and may-in principle-act as a seed field for kinematic magnetic dynamo. The correlation at later times with the above initial condition is provided by (9), which yields ij kk (ρ, 0|r, t).
For the limit κ → 0 in the KK model, the scaling relation (22) for F then implies that Thus, C (0) is a self-similar solution of ∂ t C (0) = MC (0) . It is clearly the self-similar solution with parameter α = 0 in our general classification. On the other hand, if we take A = 1/d then also The latter expression denotes the correlation of linevectors which started as the same random unit vector at time 0, at any pair of points, which ended up at time t at points displaced by r. We should emphasize that this result is valid for any divergence-free advecting velocity field and thus applies as well to incompressible fluid turbulence. It immediately follows by summing over i = j and setting r = 0 that Our analysis in the previous section can be applied to describe the behavior of C (0) . The formula C L (r, t) = Γ(r/L(t)) holds using the analytic expression (66) for Γ with α = 0 and L(t) = (D 1 t) 1/γ , valid for all r ≫ ℓ κ . It is more interesting to consider various asymptotic behaviors. The "permanence of large eddies" implies that In the convective range Finally, for r → 0 and long times, which grows with decreasing κ or increasing t, but only as a modest power law. This result may be interpreted in terms of material-line correlations by setting A = 1/d : which implies that this quantity grows slowly with time. It would be of great interest to determine the timedependence also in the dynamo regime. If the leading eigenfunction of M * satisfies d d r E kℓ (r) = 0, then R(t) need not grow exponentially. Note, for example, that the space-integral of the dual eigenfunction E ij (r) does vanish, so the issue is not straightforward.

IV. FINAL DISCUSSION
Our work leads to several important conclusions regarding the small-scale turbulent kinematic dynamo.

A. Breakdown of Flux Freezing and Dynamo
In order to understand the turbulent dynamo process a crucial fact is that magnetic field lines are not frozen into the plasma flow, even in the zero-resistance limit κ → 0. Flux-freezing would imply that only a single field line is advected into each space point from the field configuration at an earlier time. In fact, infinitely-many field lines are carried to each point by a combination of fluid advection and resistive diffusion [13,17]. In the Kraichnan velocity ensemble, the probability for two line elements to arrive at the same point at time t starting from points separated by r at time 0 is P (0, t|r, 0) ∝ exp(−r γ /γD 1 t) in the limit κ → 0 and does not degenerate into a delta function δ d (r) [16]. This is a manifestation of the phenomenon of "spontaneous stochasticity" first pointed out by Bernard et al. [16], which is due to the explosive separation of pairs of fluid particles undergoing turbulent Richardson diffusion. It was argued in [30] that this behavior as κ → 0 holds in general for a turbulent plasma with a rough velocity field, so that Alfvén's theorem on flux-conservation remains as a stochastic law only.
The breakdown of flux-freezing in the case of rough velocity fields renders the turbulent kinematic dynamo an even more subtle problem than the laminar (or large Prandtl number) kinematic dynamo (for the latter, see e.g. refs. [31,32,33,34,35].) For the very smooth velocities considered there (ξ = 2), Alfvén's theorem holds in its usual form in the limit κ → 0. However, for rougher velocities with rugosity exponent anywhere in the range 0 < ξ < 2, an infinite number of field lines enters each point even in the zero-resistance limit. The resultant magnetic field is the resistive average over the field vectors of all of the individual lines. We have shown in this work that the presence of small-scale kinematic dynamo effect depends upon the existence of sufficient angular correlation between the individual field vectors. Thus, dynamo action occurs in the KK model for smoother velocities with ξ * < ξ < 2 but not for rougher velocities with 0 < ξ < ξ * . This is true despite the fact that the stretching rate of individual fields lines is much greater for ξ smaller.
In section II C we defined R kℓ (r, t) = F ii kℓ (0, t|r, 0), which measures the correlation between line-elements δℓ k (t) and δℓ ℓ (t) at the same point at time t which started out as unit vectorsê k ,ê ℓ at distinct points separated by r at time 0. We found there that, in the nondynamo regime of the KK model with 0 < ξ < ξ * , this quantity scales as (48) for ℓ ν , ℓ κ ≪ r ≪ (D 1 t) 1/γ . Here Z is an appropriate zero-mode of M * scaling as Z kℓ (r) ∝ r ζ , with −d < ζ < 0. [Note that the factor ℓ ζ1 κ arises from W ii (0, 1) in the slow-mode expansion.] This correlation decays only as a power for r increasing through the inertial-convective range, implying that line vectors initially separated by distances ∼ (D 1 t) 1/γ contribute substantially to the final average. The correlation does not vanish as κ → 0 but, in fact, increases as a moderate power of ℓ κ , demonstrating that infinitely-many field lines continue to contribute in that limit. The result is, however, a correlation R kℓ (r, t) slowly decaying in time. On the other hand, the lengths of the individual line-elements δ 2 ℓ k (t) , δ 2 ℓ ℓ (t) grow exponentially in time as in (37) with rate λ ∝ ν/ℓ 2 ν = 1/t ν . The result is that exponentially rapidly either as t → ∞ or as κ → 0 with ν < P r c κ. We conclude that the dynamo fails for a very rough velocity field because advected line-vectors arrive at the same point with insufficient angular correlation.
Although individual field-lines are stretched to an incredible degree, resistive averaging of nearly uncorrelated lines leads to almost complete cancellation. The situation is qualitatively different in the dynamo range with smoother velocities (ξ * < ξ < 2). In that case, we have from (49) that where E 0 is the dynamo growth rate. Since E 0 ∝ 1/t κ ≪ λ ∝ 1/t ν , for λ in (38), it is still true that the angular correlations (75) decay exponentially either as t → ∞ or as κ → 0 with P r small enough. However, the decay exponent is reduced by a finite amount. Enough correlations remain between line-elements entering a point that the net magnetic field after resistive averaging can profit from stretching of individual lines and exponential growth of magnetic energy ensues.

B. Hydrodynamic and MHD turbulence
Much of the formalism of this paper carries over to the problem of kinematic dynamo for a weak seed magnetic field in hydrodynamic turbulence. The propagators F ij kℓ (ρ, t|r, 0) = F ij kℓ (r, 0|ρ, t) give a fundamental description of the kinematic dynamo for any incompressible advecting flow. All of the results of section II A apply in general, in particular equations (9), (11) and (13), and also the relationship in section III E between magnetic induction and line-vector correlations. Any further simplifications from space-homogeneity and time-stationarity also apply where appropriate. On the other hand, some features of the KK model are quite special and do not apply more generally. The self-similarity property (22) of the propagators F and F does not carry over to hydrodynamic turbulence, because of small-scale intermittency of the advecting velocity field. Also, the statistics of forward and backward Lagrangian trajectories are not identical in hydrodynamic turbulence [36]. Thus, relations such as (16) which depend upon time-reversal symmetry do not apply to the real turbulent dynamo. Lastly, the time-evolution of the propagators F and F is in general non-Markovian and thus the simple diffusion equations such as (20), (21) do not apply. One of us (G.E.) is currently carrying out a numerical evaluation of these propagators for hydrodynamic turbulence, which will be reported elsewhere.
We expect that many of the ideas of this work will apply even to nonlinear MHD turbulence and dynamo effect there. A stochastic form of flux-freezing and Alfvén's theorem holds also for non-ideal (viscous and resistive) MHD [17]. We expect these conservation laws to remain stochastic in the limit κ → 0, ν → 0 with P r fixed [30]. However, there will be nontrivial differences from the kinematic problem studied here, due to backreaction of the magnetic field on the plasma flow via the Lorentz force. For example, in comparison with hydrodynamic turbulence, 2-particle relative diffusion in MHD turbulence is observed to be suppressed in the direction transverse to the local magnetic field [37]. In principle, however, one can account for all such nonlinear effects by the presence of a second stochastic conservation law, a modified Kelvin theorem [17,38,39]. We believe that "spontaneous stochasticity" and the implied stochasticity of magnetic-line motion and flux-conservation must play a central role in the understanding of MHD turbulence, dynamo and reconnection.

V. APPENDIX: SLOW MODE EXPANSION FOR NON-HERMITIAN EVOLUTION
Unlike for the passive scalar, the n-body evolution operators M n for the passive magnetic field in the Kraichnan model, are no longer even formally Hermitian, with M * n = M n . Nevertheless, certain important properties of the scalar evolution operators remain true for M n and M * n : these are homogeneous of degree −γ, realitypreserving, and-in the non-dynamo regime-having absolutely continuous spectrum on the negative real axis. As we shall show in the following, the above properties together with assumed analyticity conditions allow us to generalize the zero-mode and slow-mode expansions derived in [16] for the Hermitian case to pairs of non-Hermitian operators M and M * . Although our intended application is to the Kazantsev-Kraichnan kinematic dynamo model, we shall carry out the derivation in an abstract, general setting. We shall employ the properties of M and M * given above and, also, other properties that will be stated explicitly below. The entire argument is modelled very closely after that in [16], with just a few important differences that are stressed below.

A. The Zero-Mode Expansion
We shall assume that the operators M and M * act on where the subscript (x or y) indicates on which variable the operator acts. Note that these Green's functions are both real-valued and, of course, G(x, y) = G(y, x).
Our aim is to derive the following short-distance asymptotic expansion for G : where * here denotes complex-conjugation. The function f a is a regular zero-mode of M with scaling dimension ζ a , while g a is a singular zero-mode of M * with scaling dimension ω a = −d + γ − ζ * a . What dominates in the expansion (77) is the contributing zero-mode whose scaling exponent ζ a has the smallest real part. Thus, we label the exponents according to the magnitude of their real part, so that Re ζ a > Re ζ a ′ and Re ω a < Re ω a ′ for a > a ′ . We derive also a similar expansion for the adjoint Green's function where now f a is a regular zero-mode of M * with scaling dimension ζ a , while g a is a singular zero-mode of M with scaling dimension ω a = −d + γ − ζ * a . We thus see that the homogeneous zero-modes of the operators M and M * come in pairs, (f a , g a ) and (f a , g a ), with related scaling exponents.
Following [16], we employ the Mellin transform, which is a unitary transformation between the spaces L 2 (R d ) and L 2 (Re σ = −d/2) ⊗ L 2 (S d−1 ) given by with the inverse transform, for R = |x|, The inner product on L 2 (Re σ = −d/2) ⊗ L 2 (S d−1 ) is: (81) However, it is more convenient to write this as (82) Re σ = −d/2, the second expression is analytic in σ wheñ f (σ,x) is analytic. This form of the inner product allows one to shift integration contours in the complex σ-plane.
A key role in the analysis is played by the operator which is homogeneous of degree zero. Since it thus commutes with the self-adjoint generator D = 1 i x·∇ x + d 2 of dilatations, it is partially diagonalized under the Mellin transform: whereÑ (σ) for each σ is an operator on L 2 (S d−1 ). Using M −1 = R γ/2 N −1 R γ/2 , one straightforwardly derives the following fundamental identity for the Green's function G(x, y) = −M −1 (x, y): See [16]. We note that the shifts in σ arise because R γ/2 acts as a translation by −γ/2 under the Mellin transform. The above identity is the key to deriving the zero-mode expansion for G.
The main hypothesis is that the operator functioñ N −1 (σ) is meromorphic in a wide vertical strip around the line Re σ = −d/2, whose only singularities are poles at complex values ζ a , a = 1, 2, . . . in the strip. By moving the integration contour in (84) further and further to the right, one picks up successive pole contributions. This implies that Green's function for large L satisfies: with the function which is homogeneous of degree ζ a in x and of degree −d + γ − ζ a in y. From the definition of the Green's function, using from which we get M x Z a (x, y) = 0 for points off the diagonal. Likewise, from which we get M * y Z a (x, y) = 0 for points off the diagonal. We finally conclude that Z a (x, y) for fixed y is a homogeneous zero-mode of M x of degree ζ a and for fixed x is a homogeneous zero-mode of M * y of degree −d + γ − ζ a . If we assume that zero-modes of a given scaling exponent are non-degenerate, as will generically be true, then we can write where f a is the unique scaling zero-mode of M with exponent ζ a and g a is the scaling zero-mode of M * with exponent ω a = −d + γ − ζ * a . We have used here the fact that M * is reality-preserving. This yields (77). The expansion (78) for G is derived by an identical argument.
Although we shall not employ the corresponding largedistance expansion in this work, we make here a few remarks about it. Under the Mellin transform the adjoint of N −1 has the kernel This last relation reveals the important fact that if N −1 (σ) has a pole at ζ a then N * −1 (σ) has a pole at ω a = −d + γ − ζ * a . Indeed we have: At this pole with Re ω a < −d/2 the role of the regular and singular zero-modes in the residue is reversed. By pushing the integration contour in (84) further and further to the left one can thus derive a large-distance expansion for G. This can also be directly obtained from the short-distance expansion for G, as follows: for L ≫ 1. Of course, a similar expansion holds for G.

B. The Slow Mode Expansion: Elementary Arguments
Define the heat-kernels so that, obviously, P (x, t|x 0 , t 0 ) = P (x 0 , t|x, t 0 ). We have the relations G(x, y) = Given the validity of the zero-mode expansions for G and G one should expect that related expansions hold for P and P . We shall show that this is indeed true, with the asymptotic expansion analogous to (77) for L ≫ 1 : (88) Here f a,p are the tower of regular slow modes of M, satisfying −Mf a,p = f a,p−1 and f a,0 = f a . Also, g a,p are solutions of ∂ t g a,p (x, t) = M * g a,p (x, t) with initial conditions g a,−1 (x, 0) = g a (x) and g a,p+1 (x, 0) = −M * g a,p (x, 0). They satisfy the scaling relations g a,p (λx, λ γ t) = λ ω a −(p+1)γ g a,p (x, t). Note that the dominant contribution in (88) will generally come from the tower with minimum Re(ζ a ) and from the first (zero-mode) term p = 0. There is an analogous expansion for P with L ≫ 1: with the roles of the operators M and M * reversed. We shall derive the above expansions in this section and the next. Here we proceed by assuming that a general expansion exists for L ≫ 1 of the form We shall then identify the form this expansion must take.
In the following section we establish from a more fundamental point of view the existence of such an expansion. First we substitute (90) into We see that whenever the asymptotic series contains a term proportional to f α (x) with scaling exponent ρ α it must also contain a term M x f α (x) with exponent ρ α −γ, and then a term M 2 x f α (x) with exponent ρ α − 2γ, and so on. This cannot continue indefinitely, since, otherwise, there would be successively more and more divergent terms for L ≫ 1. The only way that this sequence can terminate is if, eventually, M p+1 x f α (x) = 0 for some integer p. In that case, we see that f α = (−M x ) p f a ≡ f a,p for some homogeneous zero-mode f a , and the expansion (90) contains the whole tower above that zero mode. All such towers associated to regular zero modes must appear because the condition (86) together with the zero-mode expansion for G implies that The expansion (90) thus must have precisely the form of equation (88) and we must only establish the properties of g α = g a,p . We note from (91) that ∂ t g a,p = M * g a,p = −g a,p+1 .

C. The Slow Mode Expansion: Fundamental Derivation
We shall now demonstrate the existence of the expansion (90) and verify by an independent argument its general properties discussed above. A key fact that we use is that the operators M and M * both have spectrum absolutely continuous over the negative real axis. This assumption explicitly rules out kinematic dynamo effect due to point spectrum on the positive real axis. Because of this assumed property, we may define where the branch of the natural logarithm log(z) is defined with cut along the negative real axis. Furthermore, because M and M are homogeneous of degree −γ, the operators X and X * both satisfy the Heisenberg commutation relations where D is the self-adjoint generator of dilatations. We may decompose X, X * into Hermitian and skew-Hermitian parts, as where H, K are both Hermitian. In that case, we see that We can now follow the arguments in Ref. [16] to infer that under the unitary Mellin transform where Re(σ) = − d 2 . As in Ref. [16], the operators U(σ) in Eq. (98) are unitary operators on L 2 (S d−1 ) and the result in Eq. (98) follows from the Stone-von Neumann theorem on uniqueness of representations of the Heisenberg algebra. The operators K 0 (σ) in (99) are self-adjoint operators on L 2 (S d−1 ) and the result (99) is a consequence of the second half of (96)-commutativity of D and K-so that K leaves invariant the eigenspaces of D. It is convenient to introduce instead the self-adjoint operators K(σ) = U −1 (σ) K 0 (σ) U (σ). Thus, We now introduce the operators L(σ) on L 2 (S d−1 ) satisfying The operators L(σ) and L −1 (σ) can be defined explicitly by ordered exponentials along the line σ = − d 2 + iν : (104) and (105) It follows that γ∂ σ + i K(σ) = L(σ)γ∂ σ L −1 (σ), γ∂ σ − i K(σ) = L * −1 (σ)γ∂ σ L * (σ).
This is the main result that we require. The rest of the derivation of the slow mode expansion follows the argument of Ref. [16], assuming that V (σ) extends to a meromorphic operator-valued function of σ. We shall sketch here the main points. Note first that we can exponentiate the relations (108),(109) to obtain where we have defined the operators V and V * by which are mutual adjoints. From the definition N −1 = R −γ/2 M −1 R −γ/2 and (111) we see that −N −1 = (R −γ/2 V R γ/2 )(R γ/2 V −1 R −γ/2 ), which under Mellin transform becomes Of course, we have also by an identical argument. One immediate consequence of (112) is that poles of N −1 σ − γ 2 can arise only from poles of V (σ) or zeroes of V (σ − γ). Our main assumption will be that all of the poles of V (σ) lie in the half-plane Re σ > −d/2 and all of its zeroes lie in the half-plane Re σ < −d/2. Because of the adjoint relation we see that V * −1 (σ) then enjoys the same property, with the poles of V (σ) corresponding to zeroes of V * −1 (σ) and the zeroes of V (σ) corresponding to poles of V * −1 (σ).
The assumption on V implies that all of the poles of N −1 σ − γ 2 for Re σ > −d/2 + γ/2 must arise from poles of V (σ) with the form in order to reproduce the known poles of N −1 σ − γ 2 . On the other hand, we can rewrite (112) as V (σ + γp) = − N −1 σ + γ p − 1 2 V (σ + γ(p − 1)), for p = 1, 2, . . . . Let us assume that none of the poles of N −1 σ − γ 2 occur at points in the complex σ-plane with real parts differing by integer multiples of γ. This will hold generically. In that case, N −1 σ a + γ p − 1 2 is a regular operator for all p = 1, 2, . . . and we may use the above relation to infer inductively a series of poles Pushing the σ-integration contour further and further to the right gives the expansion for L ≫ 1 : (e tM ϕ)(x/L) ∼ = a,p L −σ−γp f a,p (x) g a,p (t), ϕ , with a suitable definition of g a,p (t) See [16] for more details. The above is just an integrated form of the slowmode expansion (88) for P . The slow-mode expansion (89) for P follows by an identical argument, in which the various terms arise from the poles of V * −1 (σ) in the half-plane Re σ > −d/2. There are similar large-distance expansions for P and P , in which enter the "tunnels" of singular slow modes. The terms in these expansions arise from the zeroes of V (σ ′ ) and V * −1 (σ ′ ) in the half-plane Re σ ′ < −d/2 by moving σ ′ -integration contours to the left in a formula similar to the above. The reader may work out details. We shall just note here that the pole (115) of V (σ) implies via the relation and thus the zero of V * −1 (σ) at σ = ω a − γ. This zero and the "tunnel" of zeroes beneath it give rise to the terms in the large-distance expansion of P . In our discussion throughout the appendix, we have assumed that all the regular zero modes of M and M * have scaling exponents σ with Re σ > −d/2 + γ/2 and all the singular zero modes have scaling exponents with Re σ < −d/2 − γ/2. This is true in the KK model only for d ≥ 6 and for ξ not too large. For d ≤ 4 the two primary "singular modes" have exponents ω 1 , ω 1 ≥ −d/2 for all ξ and, for sufficiently large ξ, these exponents even cross and become larger than ζ 1 , ζ 1 , respectively! For d ≥ 6 it still happens that ζ 1 < −d/2 and ω 1 > −d/2 for sufficiently large ξ. These results are not consistent with the assumptions made in the derivation sketched above. Nevertheless, the zero-and slow-mode expansions seem to hold for all d > 2 and 0 < ξ < ξ * .